# Cosmologies with positive $\lambda$: Hierarchies of future behaviour.

Helmut Friedrich
Max-Planck-Institut für Gravitationsphysik
Am Mühlenberg 1
14476 Golm, Germany

###### Abstract

Smooth Cauchy data on $\mathbb{S}^{3}$ for the Einstein-$\lambda$-vacuum field
equations with cosmological constant $\lambda>0$ that are sufficiently close
to de Sitter data develop into a solution that admits a smooth conformal
boundary ${\cal J}^{+}$ in its future. The conformal Einstein equations
determine a smooth conformal extension across ${\cal J}^{+}$ that defines on
‘the other side’ again a $\lambda$-vacuum solution. In this article we discuss
to what extent these properties generalize to the future asymptotic behaviour
of solutions to the Einstein-$\lambda$ equations with matter. We study FLRW
solutions and the Einstein-$\lambda$ equations coupled to conformally
covariant matter transport equations, to conformally privileged matter
equations, and to conformally non-covariant matter equations. We present
recent results on the Einstein-$\lambda$-perfect-fluid equations with a non-
linear asymptotic dust or asymptotic radiation equation of state.

##  1 Introduction

Roger Penrose suggested to discuss the asymptotic behaviour of space-times in
terms of extensions of their conformal structure [25], [26]. The idea is that
a physical space-time $(\hat{M},\hat{g}_{\mu\nu})$ may admit a smooth
extension $(M,g_{\mu\nu},\Omega)$ where $M$ is a smooth manifold with boundary
${\cal J}$, $g_{\mu\nu}$ a smooth Lorentz metric and $\Omega$ a smooth
function on $M$, so that $M=\hat{M}\cup{\cal J}$, $\Omega>0$ and
$g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu}$ on $\hat{M}$, while $\Omega=0$ on
the set ${\cal J}$ (referred to as the conformal boundary or short as Scri).
It may not be easy, or not possible at all, to find for a given space-time
$(\hat{M},\hat{g}_{\mu\nu})$ a smooth conformal extension
$(M,g_{\mu\nu},\Omega)$ as above, but if it can be done, the construction
provides precise and complete information about the asymptotic behaviour of
the space-time in the neighbourhood of Scri.

In the first 20 years following the introduction of the concept there became
available related general observations, various discussions of fields and
physical quantities on and near conformal boundaries, and detailed
verifications of the existence of conformal boundaries in the case of a number
of important exact solutions with symmetries. After becoming acquainted with
the idea I tried to understand to what extent the concept applied to general
solutions of Einstein’s equation

$R_{\mu\nu}[\hat{g}]-\frac{1}{2}\,R[\hat{g}]\,\hat{g}_{\mu\nu}+\lambda\,\hat{g}_{\mu\nu}=\hat{T}_{\mu\nu},$
(1.1)

with cosmological constant $\lambda>0$ and energy momentum tensor
$\hat{T}_{\mu\nu}$.

The conformal extension idea will be illustrated here by a discussion of FLRW
models because some of their features will be important for us in the
following. By these we understand space-times with manifold $\mathbb{R}\times
S$ and metric $\hat{g}=-dt^{2}+a^{2}\,\hat{h}$, where $a=a(t)$ is a scalar,
$S$ is a 3-dimensional manifold, and $\hat{h}$ a $t$-independent Riemannian
metric of constant curvature on $S$ with Ricci scalar $R[\hat{h}]=const.\geq
0$. The space-times are required to satisfy the Einstein-$\lambda$-perfect-
fluid equations with flow field $U=\partial_{t}$, total energy density
$\hat{\rho}(t)$, pressure $\hat{p}(t)$, and an equation of state
$\hat{p}=w(\hat{\rho})$.

To discuss the solutions in terms of a conformal representation we use the
rescalings

$\hat{g}_{\mu\nu}\rightarrow
g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu},\quad\quad\hat{U}^{\mu}\rightarrow
U^{\mu}=\Omega^{-1}\,\hat{U}^{\mu},\quad\quad\hat{\rho}\rightarrow\rho=\Omega^{-e}\,\hat{\rho},\quad$

with a conformal factor $\Omega=\Omega(t)$ whose evolution is fixed by the
requirements that the Ricci scalars of $g$ and $\hat{h}$ satisfy
$R[g]=R[\hat{h}]$ throughout the solution and $a\,\Omega=1$,
$\,d(a\,\Omega)/dt=0$ on a given slice $\\{t=t_{*}\\}$. The constant $e$ will
be determined later.

Then $\Omega=a^{-1}$ and in terms of the coordinate
$\tau(t)=\tau_{*}+\int^{t}_{t_{*}}\,a^{-1}\,dt$ follows

$g=-d\tau^{2}+\hat{h}.$

With a linear equation of state $w(\hat{\rho})=w_{*}\,\hat{\rho}$ where $0\leq
w_{*}=const.\leq 1/3$ the conformal analogues of the well known Friedmann and
energy conservation equations, that represent the content of (1.1), then read
with the dot denoting $d/d\tau$,

$(\dot{\Omega})^{2}=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\Omega^{e}\,\frac{\rho}{3},\quad\quad\dot{\rho}=\Omega^{-1}\,(3+3\,w_{*}-e)\,\rho\,\dot{\Omega}.$

With initial data $\Omega=\Omega_{*}=\Omega(\tau_{*})>0$ and
$\rho=\rho_{*}=\rho(\tau_{*})\geq 0$ at $\tau=\tau_{*}$ this implies

$\rho=\rho_{*}\left(\frac{\Omega}{\Omega_{*}}\right)^{3+3\,w_{*}-e},$

and thus, independent of the choice of $e$, the conformal Friedmann equation
in the form

$(\dot{\Omega})^{2}=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\frac{\rho_{*}}{3}\left(\frac{\Omega}{\Omega_{*}}\right)^{3+3\,w_{*}}.$

Let $\Omega_{*}$ be small (that is $a(t_{*})$ be large) so that the right hand
side is positive for $\Omega\leq\Omega_{*}$ and choose the sign of the square
root and the parameter $\tau$ so that $\dot{\Omega}$ is decreasing while
$\tau$ is increasing. The equation can then be integrated until
$\Omega\rightarrow 0$ at some finite value $\tau_{**}$ of the parameter. Since
$a\rightarrow\infty$ and $t\rightarrow\infty$ as $\Omega\rightarrow 0$, the
hypersurface Scri = $\\{\Omega=0\\}\sim S$ defines a boundary of the physical
space-time that represents future time-like infinity with respect to the
physical metric $\hat{g}_{\mu\nu}=\Omega^{-2}\,g_{\mu\nu}$ and that is space-
like with respect to $g_{\mu\nu}$.

That the zero of the function $\Omega$ is given here by a finite value of the
conformal coordinate $\tau$ helps to recognize subtle matter dependent
differences in the asymptotic behaviour of $a(t)$ as $t\rightarrow\infty$. In
the vacuum case $\rho_{*}=0$ we get with $t_{*}=\tau_{*}=0$, $\lambda=3$,
$S=\mathbb{S}^{3}$, and $\hat{h}=h_{\mathbb{S}^{3}}$ whence $R[\hat{h}]=6$,
the solution $\Omega=\cos\tau$. Then $t=\log\tan(\tau/2+\pi/4)$ and $a=1/\cosh
t$ which gives the maximally symmetric, geodesically complete, conformally
flat de Sitter solution

$\hat{M}=\mathbb{R}\times\mathbb{S}^{3},\quad\quad\hat{g}=-dt^{2}+\cosh^{2}t\,\hat{h}.$
(1.2)

It is here not so much important for us that the solution can be given
explicitly but that we get precise information on the asymptotic behaviour of
$\Omega(\tau)$ near Scri. The solution of the ODE above extends smoothly to
Scri and in fact beyond. While the ‘physical’ de Sitter metric is defined for
$t\in\mathbb{R}$, which is covered by $\tau$ with $-\pi/2<\tau<\pi/2$, the
cyclic function $\Omega(\tau)$ and the metric $g$ are defined and smooth for
$\tau\in\mathbb{R}$, defining a sequence of (isometric) vacuum solutions which
are separated by Scri’s.

When $\rho_{*}>0$ and $0\leq w_{*}\leq 1/3$ different cases occur. If
$0<w_{*}<1/3$ there arise smoothness and extension problems. The solutions do
approach the value $\Omega=0$ but only a few derivatives of the function
$(\Omega/\Omega_{*})^{3+3\,w_{*}}$ have a finite limit as $\Omega\rightarrow
0$. Some solutions, such as the one obtained with $w_{*}=1/9$ for instance,
admit a unique extension into a range where $\Omega<0$ which is smooth where
$\Omega\neq 0$ and drops smoothness as $\Omega\rightarrow 0$. Others, like the
one obtained for $w_{*}=1/6$, do not admit an extension beyond $\Omega=0$ as
solution to the equation above.

However, in the case of pure dust, where $w_{*}=0$, or in the case of
(incoherent) pure radiation, where $w_{*}=1/3$, the solutions extend smoothly
to $\Omega=0$ and beyond. (The unusual word pure is added here to distinguish
these cases clearly from related ones considered later). We set for
convenience $\Omega_{*}=1$, $e=3+3\,w_{*}$ and consider the case of pure dust
with $e=3$ and the case of pure radiation with $e=4$. We have then
$\rho=\rho_{*}=const.>0$ and get the conformal Friedmann equation in the form

$(\dot{\Omega})^{2}=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\frac{\rho_{*}}{3}\,\Omega^{e}.$

The global solutions and the solution manifold look in these two cases as
follows.

Pure dust solutions. Depending on the real roots of the polynomial

$P(\Omega)=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\frac{\rho_{*}}{3}\,\Omega^{3},$
(1.3)

three cases occur (assuming in the following discussions suitable parameters
$\tau$ and signs of the square root of $P$).

(i) $R[\hat{h}]^{3}>54\,\lambda\,\rho_{*}^{2}$: $P$ has roots
$\Omega_{-}<0<\Omega_{1}<\Omega_{2}$. There are solutions that take values in
$[\Omega_{2},\infty[$ start from a Big Bang (where $\Omega\rightarrow\infty$,
resp. $a\rightarrow 0$ for a finite value of the parameter), decrease, reach
their minimum $\Omega_{2}$ (i.e. $a$ reaches a maximum), increase again and
approach a Big Crunch where $\Omega\rightarrow\infty$ for a finite value of
the parameter.

The solutions that take values in $[\Omega_{-},\Omega_{1}[$ are cyclic,
oscillating between $\Omega_{1}$ and $\Omega_{-}$ and passing through Scri’s
on the way. These approach the de Sitter solution as $\rho_{*}\rightarrow 0$.

(ii) $R[\hat{h}]^{3}=54\,\lambda\,\rho_{*}^{2}$: Then
$P=\left(\Omega-\frac{R[\hat{h}]}{3\,\rho_{*}}\right)^{2}\left(\Omega+\frac{R[\hat{h}]}{18}\right)$
with simple root $\Omega_{-}=-\frac{R[\hat{h}]}{18}$ and double root
$\Omega_{+}=\frac{R[\hat{h}]}{3\,\rho_{*}}$. The solutions that take values in
the domain $]\Omega_{+},\infty[$ start from a Big Bang, decrease monotonously
and approach the value $\Omega_{+}$ without ever assuming it. Similarly,
starting from their minimum value $\Omega_{-}$ the solutions which take values
in $[\Omega_{-},\Omega_{+}[$ are increasing, pass Scri’s, and approach
asymptotically the value $\Omega_{+}$ in both directions. Finally there is the
time independent solution $\Omega(\tau)=\frac{R[\hat{h}]}{3\,\rho_{*}}$.

iii) $0\leq R[\hat{h}]^{3}<54\,\lambda\,\rho_{*}^{2}$: $P$ has one real root
$\Omega=\Omega_{-}<0$. The solutions start with $\Omega>0$ from a Big Bang,
decrease monotonously, pass through a Scri for a finite value of the
parameter, become negative, assume their minimum $\Omega_{-}$, increase again,
pass through another Scri and approach a Big Crunch where
$\Omega\rightarrow\infty$.

This solution admits a smooth cyclic extension in the following sense. Before
the limit $\Omega\rightarrow\infty$ is achieved the function
$\omega=\Omega^{-1/2}$ is defined and satisfies

$4\,(\dot{\omega})^{2}=\frac{\rho_{*}}{3}-\frac{R[\hat{h}]}{6}\,\omega^{2}+\frac{\lambda}{3}\,\omega^{6}.$
(1.4)

With a redefinition of the constants this becomes the conformal Friedmann
equation above with $w_{*}=1$ (stiff matter equation of state) and $\lambda$
and $\rho_{*}$ interchanged. The condition on $R[\hat{h}]$ above ensures that
the polynomial in $\omega$ on the right hand side is positive everywhere and
the equation can be integrated across $\omega=0$. Where $\omega<0$ the
transformation $\Omega=\omega^{-2}$ connects to a second copy of the solution
above and the process can be repeated. If the parameter $\tau$ is chosen so
that $\Omega(0)=\Omega_{-}$, whence $\Omega(\tau)=\Omega(-\tau)$, the solution
is given in terms of Jacobi’s elliptic function $cn(u,k)$ [20] by

$\Omega(\tau)=\Omega_{-}+\Sigma\,\frac{1-cn(u(\tau),k)}{1+cn(u(\tau),k)},$
(1.5)

where $u(\tau)=(\rho_{*}\,\Sigma/3)^{1/2}\,\tau$, the modulus is
$k=(R[\hat{h}]+2\,\rho_{*}\,\Sigma-4\,\rho_{*}\,\Omega_{-})^{1/2}\,(8\,\rho_{*}\,\Sigma)^{-1/2}$,
$\Sigma=(3\,\Omega_{-}^{2}-\Omega_{-}\,R[\hat{h}]/\rho_{*})^{1/2}$, and the
root $\Omega_{-}$ of $P$, that satisfies $\Omega_{-}<-R[\hat{h}]/6\,\rho_{*}$,
is related to $\lambda$ by
$\lambda-R[\hat{h}]\,\Omega_{-}^{2}/2+\rho_{*}\,\Omega_{-}^{3}=0$.

Pure radiation solutions. Depending on the real roots of the polynomial

$Q(\Omega)=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\frac{\rho_{*}}{3}\,\Omega^{4},$
(1.6)

the following cases occur.

(i) $R[\hat{h}]^{2}>16\,\rho_{*}\,\lambda$: Q has four simple roots
$\Omega_{1}<\Omega_{2}<0<\Omega_{3}<\Omega_{4}$. There are solutions that take
values in $[\Omega_{4},\infty[$, start from a Big Bang, achieve their minimum
$\Omega_{4}$, increase and end in a Big Crunch. There are similar solutions
that take values in $]-\infty,\Omega_{1}]$.

There are cyclic solutions that take values in $[\Omega_{2},\Omega_{3}]$,
oscillate between $\Omega_{2}$ and $\Omega_{3}$, and pass through Scri’s on
the way. These solutions approach the de Sitter solution as
$\rho_{*}\rightarrow 0$.

(ii) $R[\hat{h}]^{2}=16\,\rho_{*}\,\lambda$: Then
$Q=\frac{\rho_{*}}{3}\left(\Omega^{2}-\frac{R[\hat{h}]}{4\,\rho_{*}}\right)^{2}$
with the two double roots
$\Omega_{\pm}=\pm\sqrt{\frac{R[\hat{h}]}{4\,\rho_{*}}}$. There is a strictly
monotonous solution which takes values in $]\Omega_{-},\Omega_{+}[$, passes
through a Scri, and approaches the values $\Omega_{-}$ and $\Omega_{+}$
asymptotically. There are two strictly monotonous solutions that take values
in $]\Omega_{+},\infty[$ and in $]-\infty,\Omega_{-}[$ respectively. The first
one approaches at one end the value $\Omega_{+}$ asymptotically and at the
other end a Big Bang. The second solution is similar. Finally there are the
time independent solutions
$\Omega(\tau)=\pm\sqrt{\frac{R[\hat{h}]}{4\,\rho_{*}}}$.

(iii) $0\leq R[\hat{h}]^{2}<16\,\rho_{*}\,\lambda$: $Q$ has no real root. The
solution takes values in $]-\infty,\infty[$. It starts from a Big Bang,
decreases with $\Omega>0$ monotonously, passes through a Scri, decreases
further, and reaches a Big Crunch where $\Omega\rightarrow-\infty$.

Again, this solution admits a smooth cyclic extension in the following sense.
With $\omega=-\Omega^{-1}$ close to the Big Crunch the equation for $\Omega$
gives

$(\dot{\omega})^{2}=\frac{\lambda}{3}\,\omega^{4}-\frac{R[\hat{h}]}{6}\,\omega^{2}+\frac{\rho_{*}}{3},$
(1.7)

which is the original equation with the roles of $\lambda$ and $\rho_{*}$
swapped. It can be integrated beyond $\omega=0$ and with $\Omega=\omega^{-1}$
be connected to a second copy of the solution for $\Omega$.

If the parameter $\tau$ is chosen so that $\Omega(0)=0$ and $\Omega(\tau)$ is
increasing with increasing $\tau$ near $\tau=0$, the solution is given in
terms of Jacobi’s elliptic functions by

$\Omega(\tau)=\sqrt{f}\,\,\,\frac{sn(u(\tau),k)}{cn(u(\tau),k)}\,\,\frac{k^{\prime}+dn(u(\tau),k)}{1+dn(u(\tau),k)},$
(1.8)

where $u(\tau)=\sqrt{\rho_{*}/3}\,(\sqrt{f}+e)\,\tau$,
$f=\sqrt{\lambda/\rho_{*}}$, $e=\sqrt{f/2+R[\hat{h}]/8\,\rho_{*}}$,
$k^{\prime}=\sqrt{1-k^{2}}$, $k=2\,(e+\sqrt{f})^{-1}\,\sqrt{e\,\sqrt{f}}$.

The ‘fine tuned’ cases (ii) will not be of interest to us in the following. We
shall mainly be interested in the cases (iii) where a Big Bang in the finite
past (in terms of physical time) is connected with a space-like Scri in the
infinite future. These cases admit the limits $R[\hat{h}]\rightarrow 0$ which
are in agreement with the 2018 results of the Planck team. We shall later
focus on the ends at future time-like infinity where the solutions approach
Scri.

Of course, in the standard interpretation of GR only a maximal connected
space-time with $\Omega>0$ will be considered as a physical solution. Its
conformal extension may just be considered a fun game which works because the
solutions are conformally flat and extremely simple. Heeding Tolman’s
admonition [35], which reads (with a slight variation) ‘$\ldots$ we study FLRW
models primarily in order to secure definite and relatively simple
mathematical problems, rather than to secure a correspondence with known
reality $\ldots$’ we wonder: Do any of the observations above extend to more
general situations ? We are in particular interested in examples where the
fluid flow is not forced to be geodesic and which are not conformally flat so
as to admit gravitational radiation, a concept that FLRW models allow to talk
about only in terms of approximations.

Answers of any generality to the question above need global or semi-global
results on suitable Cauchy problems for Einstein’s field equations. Moreover,
as shown by the subtleties discussed above, they require sharp control on the
asymptotic behaviour of the solutions at least at future time-like infinity.
At the time when Penrose put forward his proposal no such results were
available. Until the early 1980’s the understanding of the Cauchy problem for
Einstein’s equations was restricted to existence results local in time.

##  2 Conformal field equations, stability results.

Finding initial data which develop into solutions to the Einstein equations
that admit smooth conformal boundaries poses subtle problems when the
cosmological constant $\lambda$ vanishes or is negative. Because the initial
slices are not compact there have to be made choices about the fall-off
behaviour of the data (see [11], [13]).

In the same article where Einstein introduced the cosmological constant, he
came to the conclusion that cosmological solutions should be spatially compact
[3]. In fact, in contrast to the FLRW solutions, which are defined by ODE’s,
there can be no other choice if one whishes to construct geodesically null and
time-like future complete solutions to the full Einstein equations. There are
no natural boundary conditions for the evolution equations if $\lambda>0$.

As indicated above, the Planck team found the Universe to be constrained to be
spatially flat to extremely high precision [2]. Because no restrictions are
given on the ‘size’ of $S$ (in terms of the distances function defined by the
prescribed metric $\hat{h}$), we can assume $S$, which we may wish to be
simply connected, to be as large as we like so that $R[\hat{h}]$ is lying in
the error margin given in [2] while still being positive.

If the initial slice is compact, which will be assumed in the following, only
smoothness and smallness conditions can be imposed on ‘general’ Cauchy data.
The following global non-linear stability result holds [5], [6].

On a slice $S=\\{t=const.\\}\sim\mathbb{S}^{3}$ of the de Sitter solution to
Einstein’s field equations (1.1) with $\lambda=3$ and $\hat{T}_{\mu\nu}=0$
consider smooth Cauchy data for these equations. If these data are (in terms
of suitable Sobolev norms) sufficiently close to the de Sitter data on $S$,
they develop into solutions to (1.1) that are time-like and null geodesically
complete and admit smooth space-like conformal boundaries at past and future
time-like infinity.

Because the cosmological constant can be given any positive value by a
conformal rescaling with a constant conformal factor, the precise value of
$\lambda$ is irrelevant here.

The technical basis of this result is a remarkable feature of the Einstein
equations. While they are designed to determine a metric, they can be
represented in terms of a conformal factor $\Omega$, the conformal metric
$g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu}$ and certain tensor fields derived
from them so that they imply with suitable gauge conditions equations that can
be hyperbolic even where the conformal factor vanishes or becomes negative,
i.e. beyond the domain where $\hat{g}_{\mu\nu}$ is defined [4]. We refer to
these equations as conformal Einstein equations. It should be noted that this
name has subsequently also been used for conformal representations of the
Einstein equations which were derived for other purposes and do not share the
properties used below.

Conformal de Sitter space, given above by
$M=]-\pi/2,\pi/2[\times\mathbb{S}^{3}$, $\,g=-d\tau^{2}+\hat{h}$,
$\,\Omega=\cos\tau$, is a solution to the conformal Einstein equations that
extends smoothly, as a solution to the equations and with the same expressions
for the metric and conformal factor, beyond the boundaries
$\\{\tau=\pm\pi/2\\}$ to all of $\mathbb{R}\times\mathbb{S}^{3}$. Then
$\Omega<0$ on the slices $\\{\tau=\pm\pi\\}$. Consider Cauchy data $\bar{g}$,
$\bar{\Omega}$, $\ldots$ for the conformal field equations on a slice
$\\{\tau=\tau_{*}\\}$ with $|\tau_{*}|<\pi/2$ which are ‘general’ in the sense
that symmetries are not necessarily imposed. If these data are sufficiently
close to the conformal de Sitter data induced on $\\{\tau=\tau_{*}\\}$,
general properties of hyperbolic equations [19] guarantee that the solution
$\bar{g}$, $\bar{\Omega}$, $\dots$ to the conformal field equations which
develop from the general data also exist on the domain $\\{|\tau|\leq\pi\\}$
and the conformal factor $\bar{\Omega}$ is negative on $\\{\tau=\pm\pi\\}$.
The conformal field equations then ensure that there exist two hypersurfaces
${\cal J}^{\pm}\subset\\{|\tau|\leq\pi\\}$ with $\bar{\Omega}|_{{\cal
J}^{\pm}}=0$, $d\bar{\Omega}|_{{\cal J}^{\pm}}\neq 0$ that are space-like with
respect to $\bar{g}$ and sandwich a domain $\hat{M}\subset\\{|\tau|\leq\pi\\}$
on which $\bar{\Omega}>0$. Then $(\hat{M},\hat{g}=\bar{\Omega}^{-2}\,\bar{g})$
is the desired solution to the Einstein equations.

The fact that the set of all Cauchy data on $\mathbb{S}^{3}$ for the Einstein
vacuum equations with positive cosmological constant contains an open subset
(in terms of suitable Sobolev norms) of data which develop into solutions that
are conformally well behaved at future and past time-like infinity shows that
the existence of smooth conformal boundaries can be a fairly general feature
of solutions to Einstein’s field equations. Besides a smallness condition
there are no restrictions on the conformal Weyl tensor.

Recovering from the technical struggles that led to this insight, I began to
wonder:

If the field equations ensure a smooth future evolution of $\Omega$ and $g$
beyond ${\cal J}^{+}=\\{\Omega=0\\}$, so that they define in the future of
${\cal J}^{+}$ another ‘physical’ solution to Einstein’s field equations with
metric $\hat{g}=\Omega^{-2}\,g$, and any gravitational radiation, represented
by non-linear perturbations of the conformal Weyl tensor, travels unimpeded
across ${\cal J}^{+}$ into that domain, why should physics come to an end at
the future conformal boundary ${\cal J}^{+}$ ?

This behaviour may be considered as just another quirk of the field equations,
that should not be taken too seriously. The history of General Relativity
shows, however, that Einstein’s equations were often wiser than their solvers.
Physicists made sense of the more exotic features of the solutions found by
Schwarzschild and Friedmann only years after their discovery.

Though it can all be found in the article referred to above, I never
explicitly speculated about this in public. Being a beginner, it would hardly
have been taken seriously, in particular, because I had no answers to the
questions:

$-$ What happens if there is matter around ?

$-$ How will matter behave in the far future ?

Our discussion of the FLRW solutions above have shown that different matter
models, exemplified there by $\rho_{*}$ and $w_{*}$, may have quite diverse
consequences. Moreover,

$-$ What could be the meaning of the solution ‘on the other side’ of ${\cal
J}^{+}$ ?

In the stability result above the space-time defined by the metric
$\hat{g}=\Omega^{-2}\,g$ on the other side of ${\cal J}^{+}$ looks like a time
reversed version of the space-time end ‘on this side’: from being infinitely
extended at ${\cal J}^{+}$ its space sections begin to shrink. This is
certainly quite different from our present idea about the beginning of a
cosmological space-time. So I kept returning to the first two questions over
the following years.

The first stability result generalizing the one above concerns the (in four
space-time dimensions) conformally invariant Maxwell- or Yang-Mills-equations
[7]. It shows:

The nonlinear vacuum stability result outlined above generalizes to the
coupled Einstein-$\lambda$-Maxwell-Yang-Mills equations. The perturbed
solutions admit smooth conformal boundaries in the future and the past. The
conformal field equations determine smooth conformal extensions of the
solutions beyond these boundaries.

This result establishes a pattern for analyzing various other situations in
which conformally covariant matter transport equations are coupled to the
Einstein-$\lambda$ equation. Christian Lübbe and Juan Valiente Kroon studied
the Einstein-$\lambda$-perfect-fluid equations with the equation of state
$\hat{p}=1/3\,\hat{\rho}$ for pure (incoherent) radiation. These matter
equations have in common with the Maxwell equations that the energy-momentum
tensor is trace-free and the conformal matter equations have the same form as
the ‘physical’ version. They show [22]:

The FLRW-solutions with the equations of state of pure radiation and a smooth
conformal boundary in the future are non-linearly future stable in the class
of all Einstein-$\lambda$-perfect-fluid solutions with this equation of state.
The perturbed solutions admit a smooth conformal boundary in the future and a
smooth conformal extension beyond.

A further example leading to a similar result is given by the Einstein
equations coupled to the massless Vlasov matter equations [18].

In the FLRW models given by (1.5), (1.8) and the generalizations discussed
above, forward Scri’s and backward Scri’s as well as Big Bangs and Big
Crunches stand back to back in the conformal extensions. Motivated by the
observations above, results on Paul Tod’s ideas about isotropic singularities
[23], [32], [33], [34], where the initial singularity is represented after a
suitable conformal rescaling by a finite space-like set similar to a ${\cal
J}^{+}$, and by his thoughts about the nature of entropy near the big bang,
Roger Penrose proposed a cosmological model, referred to as Conformal Cyclic
Cosmology (CCC) [27]. It considers a chain of universes where a given
universe, say $U_{n}$, develops in its future a well defined ${\cal J}^{+}$,
referred to now as the crossover surface, which is followed by another
universe, $U_{n+1}$, for which the crossover surface represents an isotropic
singularity. The solutions (1.5), (1.8) can be used to create such situations.

This again gives rise to complicated questions. Strongly simplifying
assumptions on $U_{n}$ and $U_{n+1}$ may provide situations where some kind of
identification or glueing of the different ends lead to a picture as outlined
above. It is not clear, however, that anything similar can be done with any
degree of control if more general, conformally curved, solutions to Einstein’s
equations are considered. The precise nature of the transition from the
infinite future of $U_{n}$ to the beginning of $U_{n+1}$ is unresolved so far.
It should be brokered by a mechanism which guarantees a unique extension
without any interference from outside, but not necessarily preserving the time
reflection invariance of hyperbolic equations. This requires a closer look at
the equations and the matter models from both sides of the crossover surface,
generalizing perhaps of the work initiated by Alain Bachelot [1].

##  3 Conformally non-covariant matter fields.

Finding such a mechanism, if something like it exists at all, requires among
other things a sufficiently general and deep understanding of the behaviour of
matter and the field equations at future time-like infinity. A number of
authors analysed the future asymptotic behaviour of solutions to the
Einstein-$\lambda$-perfect fluid equations with an homentropic flow, where the
entropy is constant in space and time and the equation of state can be given
in the form $\hat{p}=w(\hat{\rho})$ with some suitable function $w$. Often
they assume a linear equation of state $\hat{p}=w_{*}\,\hat{\rho}$,
$w_{*}=const.$, and the additional condition $0<w_{*}<1/3$, see [17], [24],
[29], [30], [31]. In the articles [21] and [29] is studied the future
stability of FLRW space-times for more general classes of equations of state
$\hat{p}=w(\hat{\rho})$. All of this work was done in more conventional
representations of the field equations in terms of which the questions of
interest here may indeed be difficult to analyse. None of the authors above
looked at these things in the way indicated above.

In the following I tried to generalize the kind of analysis begun above,
hoping to answer the following question:

Can there be achieved $C^{\infty}$, $C^{k}$, $\ldots$, or at least in some
sense uniquely extendible conformal structures at future time-like infinity
for solutions to Einstein-$\lambda$-matter equations with conformally non-
covariant matter field equations ?

###  3.1 Scalar fields.

There are two results pointing into that direction. In [28] H. Ringström
studied the future stability of a very general class Einstein-non-linear
scalar field systems with a scalar field equation of the form

$\hat{\nabla}_{\mu}\hat{\nabla}^{\mu}\phi-\left(m^{2}\,\phi+V^{\prime}(\phi)\right)=0,$
(3.1)

where ${}^{\prime}=\partial/\partial\phi$, and an energy momentum tensor

$\hat{T}_{\mu\nu}=\hat{\nabla}_{\mu}\phi\,\hat{\nabla}_{\nu}\phi-\left(\frac{1}{2}(\hat{\nabla}_{\rho}\phi\,\hat{\nabla}^{\rho}\phi+m^{2}\,\phi^{2})+V(\phi)\right)\hat{g}_{\mu\nu},$
(3.2)

with a potential of the form $V(\phi)=\phi^{3}\,\mu+\phi^{4}\,U(\phi)$ where
$\mu$ is a constant and $U$ a smooth real-valued function. In [10] has been
considered a special case with the following result.

If $\mu=0$ and $3\,m^{2}=2\,\lambda$, the coupled Einstein-$\lambda$-scalar-
field equations (1.1), (3.1), (3.2) imply a reduced system of conformal field
equations for the unknowns $\Omega$,
$g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu}$, $\psi=\Omega^{-1}\,\phi$ and some
tensor fields derived from them. In a suitable gauge this is hyperbolic for
any sign of $\Omega$. Smooth Cauchy data for this system can be prescribed
with $\Omega=0$ on a compact $g$-space-like 3-manifold ${\cal J}^{+}$. The
development of these data backwards in time is smooth and induces on a space-
like slice $S$ (in the corresponding physical space-time) smooth standard
Cauchy data $\Delta_{0}$ for the coupled Einstein-$\lambda$-scalar-field
system. For $\Delta_{0}$ there exist (in terms of Sobolev norms) an open
neighbourhood of Cauchy data on $S$ for this system so that the future
development of any smooth data in this neighbourhood admit a smooth conformal
extension beyond the respective future time-like infinity.

We recall that the conformally covariant scalar operator $\phi\rightarrow
L_{\hat{g}}\phi=(\Box_{\hat{g}}-\frac{1}{6}\,R[\hat{g}])\phi$ with
$\Box_{\hat{g}}=\hat{\nabla}_{\mu}\hat{\nabla}^{\mu}$ satisfies
$L_{\Omega^{2}\hat{g}}(\Omega^{-1}\phi)=\Omega^{-3}\,L_{\hat{g}}\phi$. With
the trace of the energy momentum tensor (3.2) given by

$\hat{T}=-\hat{\nabla}_{\mu}\phi\,\hat{\nabla}^{\mu}\phi-2\,m^{2}\,\phi^{2}-4\,V(\phi)=-R[\hat{g}]+4\,\lambda,$

equation (3.1) can be expressed in terms of the conformally covariant wave
operator. It reads then

$L_{\hat{g}}\phi=\left(m^{2}-\frac{2}{3}\,\lambda\right)\phi+V^{\prime}(\phi)-\frac{2}{3}\,\phi\,V(\phi)-\frac{1}{6}\,\phi\,\hat{\nabla}_{\mu}\phi\,\hat{\nabla}^{\mu}\phi-\frac{1}{3}\,m^{2}\,\phi^{3},$

while we get in terms of the conformal fields $\psi=\Omega^{-1}\,\phi$ and
$g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu}$ the conformal representation

$L_{g}\psi=\Omega^{-2}\left(m^{2}-\frac{2}{3}\,\lambda\right)\psi+\Omega^{-3}\,V^{\prime}(\Omega\,\psi)-\frac{2}{3}\,\Omega^{-2}\,\psi\,V(\Omega\,\psi)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{1}{6}\,\psi\,\nabla_{\mu}(\Omega\,\psi)\,\nabla^{\mu}(\Omega\,\psi)-\frac{1}{3}\,m^{2}\,\psi^{3}.$

Equation (3.1) is thus not conformally covariant with our conditions but
conformally regular in the sense that there will remain no $\Omega^{-1}$ terms
on the right hand side if the conditions of the theorem above are taken into
account. This is a property it shares with the Einstein-$\lambda$ vacuum field
equations.

The energy momentum tensor is not trace free but satisfies

$\hat{T}=-\Omega^{2}\left(\nabla_{\mu}(\Omega\,\psi)\,\nabla^{\mu}(\Omega\,\psi)+\frac{4}{3}\lambda\,\psi^{2}+4\,\Omega^{-2}\,V(\Omega\,\psi)\right)\rightarrow
0\,\,\,\mbox{as}\,\,\,\Omega\rightarrow 0,$

where the limit above will follow only if $g_{\mu\nu}$ and $\psi$ can be shown
to extend smoothly as $\Omega\rightarrow 0$. It should be mentioned that the
hyperbolic reduced conformal field equations considered above as well as those
considered in the following preserve the constraints implied by the conformal
field equations. In the case of the present system, the proof of this fact is
far from immediate.

###  3.2 Pure dust cosmologies.

The flow fields $\hat{U}$ of Einstein-$\lambda$-perfect-fluid solutions with
pure dust equation of state

$\hat{p}=0,$

arising from data on a compact 3-manifold $S$ where $\hat{\rho}>0$ are often
interpreted as representing the cosmic flow of galaxies. The field equations
imply that the trace of the corresponding energy momentum tensor satisfies

$\hat{T}=-\hat{\rho}\neq 0.$

In the case of the conformally flat FLRW models we have seen that such
solutions can develop smooth conformal boundaries. Whether this is also
possible if the conformal Weyl tensor does not vanish is not obvious (see
[17]). In [12] it has been shown:

Consider a FLRW solution to the Einstein-$\lambda$-perfect-fluid equation with
pure dust equation of state arising from data on a Cauchy hypersurface
$S\sim\mathbb{S}^{3}$. If it admits a smooth conformal boundary at future
time-like infinity, then any smooth general set of Cauchy data on $S$ for the
same equation which is sufficiently close (in terms of Sobolev norms) to the
FLRW-data develops into a solution that is time-like and null geodesically
future complete, admits a smooth conformal boundary in its infinite future,
where $\Omega\rightarrow 0$, and extends as a smooth solution to the conformal
field equations into a domain where $\Omega<0$.

The unknowns $g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu}$ and
$\rho=\Omega^{-3}\,\hat{\rho}$ in the conformal field equations then remain
bounded as $\Omega\rightarrow 0$ and it follows that

$\hat{T}=-\Omega^{3}\rho\rightarrow 0\quad\mbox{as }\quad\Omega\rightarrow 0.$

Assuming initial data so that $\hat{\rho}>0$, the equation for the flow vector
field $\hat{U}^{\mu}$ reduces in the case of pure dust to the equation

$\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\hat{U}^{\nu}=0,$

and thus to an ODE. In terms of the conformal fields it takes the form

$0=U^{\mu}\,\nabla_{\mu}\,U_{\nu}-\Omega^{-1}\,(g^{\mu}\,_{\nu}+U^{\mu}\,U_{\nu})\nabla_{\mu}\Omega,$

and the $\Omega^{-1}$ term, which reflects the conformal non-covariance of the
system, spreads into other equations of the conformal system.

The fact that the flow is geodesic combined with the particular structure of
the energy momentum tensor of a perfect fluid allows us to make in this
particular case contact with and exploit some conformally invariant structure.

Let $\hat{g}$, $\hat{U}$, $\hat{\rho}>0$ satisfy the Einstein-$\lambda$-pure-
dust equations so that $\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\hat{U}^{\nu}=0$. If
the functions $r$ and $q$ satisfy

$0\neq\hat{\nabla}_{\hat{U}}r=q\,r,\quad\quad
0\neq\hat{\nabla}_{\hat{U}}q=-\frac{1}{2}\,q^{2}+(\lambda/6-\hat{\rho}/3)\,r,$

the curve $x(s)$ with $\frac{d}{ds}x=V=r\,\hat{U}$ and the 1-form
$b=\hat{g}(q\,\hat{U},\,\cdot\,)$ satisfy the conformal geodesic equations

$\hat{\nabla}_{V}V+2\,V<b,V>-\,\hat{g}(V,V)\,b^{\\#}=0,$
$\hat{\nabla}_{V}b\,-<b,V>b+\frac{1}{2}\,\hat{g}(V,\,\cdot\,)\,\hat{g}^{\\#}(b,b)=\hat{L}(V,\,\cdot\,),$

where $\hat{L}$ denotes the Schouten-tensor of $\hat{g}$. The point of this
observation is that the conformal geodesic equations are conformally invariant
and the conformal geodesics are invariants of the conformal structure [9].
This was used in [12] to regularize the conformal Einstein-$\lambda$-pure-dust
system near future time-like infinity.

The pure dust model is thus still conformally privileged.

###  3.3 Equations of state with prescribed asymptotic behaviour.

In the articles cited above that study the future behaviour of cosmological
solutions to the Einstein-$\lambda$-perfect fluid equations on the basis of
linear equations of state, $\hat{p}=w_{*}\,\hat{\rho}$, $w_{*}=const$, no
arguments are given for this choice. It appears to be rather a matter of
convenience instead of being motivated by a deep understanding of its physical
role. While solutions to the Einstein-$\lambda$-perfect-fluid equations may
provide good cosmological models on large scales, it seems fairly unlikely
that linear equations of state represent natural requisites from the Big Bang
to future time-like infinity. Models of the universe that behave at late times
like a pure dust or a pure radiation FLRW model or one of their conformally
curved generalizations may require transitions of the form

$\hat{p}=w^{**}(\hat{\rho})\,\hat{\rho},$

with a function $w^{**}(\hat{\rho})$ that satisfy, consistent with our earlier
requirements, $0\leq w^{**}(\hat{\rho})\leq 1/3$ and assumes the value
$w^{**}(\hat{\rho})=0$ or $w^{**}(\hat{\rho})=1/3$ at late times. There is
nothing, however, which would fix a notion of ‘late time’. The only meaningful
requirement would be that these values are approximated in the limit when the
space-time approaches future time-like infinity. The equation of state would
then still need to recognize, however, where and when this limit will be
achieved.

In the cases $(iii)$ of the FLRW models discussed above the physical density
$\hat{\rho}$ and the conformal density $\rho$ satisfy a relation of the form

$\hat{\rho}=\Omega^{e}\rho.$ (3.3)

In those cases we had $\rho=\rho_{*}=const.>0$, $e=const.>0$ whence
$\hat{\rho}\rightarrow 0$ as $\Omega\rightarrow 0$ and
$\hat{\rho}\rightarrow\infty$ as $\Omega\rightarrow\infty$. The behaviour of
$\hat{\rho}$ can thus be understood as an indicator for the approach to the
infinite future or the Big Bang. In generalizing the situation we shall keep
(3.3), hoping that it will serve the indicator function at least in the far
future where $\Omega\rightarrow 0$, but we may have to give up the relation
$\rho=\rho_{*}=const.$ We could think of generalizing (3.3) by assuming $e$ to
be a function. In this article we will only consider the situations near the
end where $\Omega\rightarrow 0$ and try to keep (3.3) with $e=const.>0$. This
will work well as long as $\rho$ can be guaranteed to stay positive and
bounded as $\Omega\rightarrow 0$.

The pure dust and the pure radiation equations of state are now generalized as
follows. An asymptotic dust equation of state is given by a function of the
form

$\hat{p}=w(\hat{\rho})=\left(\hat{\rho}^{k}\,w^{*}(\hat{\rho})\right)\hat{\rho}\,\quad\mbox{with
some}\quad k\in\mathbb{N},$ (3.4)

combined with (3.3) where $e=3$. It implies with the notation
${}^{\prime}=\partial/\partial\hat{\rho}$

$w^{\prime}(\hat{\rho})=\hat{\rho}^{k}\,\left\\{(1+k)\,w^{*}+\hat{\rho}\,\,(w^{*})^{\prime}\right\\}.$
(3.5)

An asymptotic radiation equation of state is given by a function of the form

$\hat{p}=w(\hat{\rho})=\left(\frac{1}{3}-\hat{\rho}^{k}\,\,w^{*}(\hat{\rho})\right)\hat{\rho}\quad\mbox{with
some}\quad k\in\mathbb{N},$ (3.6)

combined with (3.3) where $e=4$. It implies

$w^{\prime}(\hat{\rho})=\frac{1}{3}-\hat{\rho}^{k}\left\\{(1+k)\,w^{*}+\hat{\rho}\,(w^{*})^{\prime}\right\\}.$
(3.7)

In both case $w^{*}(\hat{\rho})$ is assumed to be a smooth function defined
for all values of $\hat{\rho}$ that satisfies

$0<w^{*}(\hat{\rho})<c=const.$

The positivity is required to clearly distinguish the asymptotic from the pure
cases. Limits $w^{*}(\hat{\rho})\rightarrow 0$ give back the pure dust and the
pure radiation equations of state.

The factors $\hat{\rho}^{k}$ with positive $k$ have been included in the
definitions as a simple means to control the speed at which the pure dust or
the pure radiation situations is approximated as $\hat{\rho}\rightarrow 0$.

The conditions on $w^{*}$ may appear crude but they suffice for analysing the
effect of the intended modifications of the equations of state in domains
where $\hat{\rho}$ becomes small. For $\hat{\rho}$ positive but sufficiently
close to zero, the range we are interested in, the terms in curly brackets in
(3.5) and (3.7) are positive. It follows that in the case of asymptotic dust
the speed of sound $w^{\prime}(\hat{\rho})$ is positive as $\hat{\rho}>0$ and
$w^{\prime}(\hat{\rho})\rightarrow 0$ as $\hat{\rho}\rightarrow 0$. In the
case of asymptotic radiation holds $w^{\prime}(\hat{\rho})>0$ for
$\hat{\rho}\geq 0$ and $w^{\prime}(\hat{\rho})$ will remain positive if the
solution can be smoothly extended into a domain where $\Omega<0$.

We note that the Cauchy problems local in time for Einstein-$\lambda$-perfect
fluids with asymptotic dust or radiation equations of state pose no problems
where $\hat{\rho}$ is sufficiently small. This follows from the results of
[8], [16] where only weak conditions on the equation of state are assumed.

The principal parts of the matter equations are affected by the equations of
state above with the consequence that any conformal covariance or privilege is
lost. Definition (3.4) implies

$\hat{T}=\hat{g}^{\mu\nu}\,\hat{T}_{\mu\nu}=3\,w(\hat{\rho})-\hat{\rho}=-\Omega^{3}\,\rho+3\,\Omega^{3+3\,k}\,\rho^{1+k}\,w^{*}(\Omega^{3}\,\rho),$

while definition (3.6) gives

$\hat{T}=-3\,\Omega^{\,4+4\,k}\,\rho^{1+k}\,w^{*}(\Omega^{4}\,\rho).$

In both case $\hat{T}\neq 0$ if $\rho>0$ and $\hat{T}\rightarrow 0$ as
$\Omega\rightarrow 0$ if $\rho$ remains bounded in this limit.  As seen below,
this last condition will be met in the case of an asymptotic radiation
equation of state while it is not clear whether this can be guaranteed also in
the case of an asymptotic dust equation of state.

The conditions on the admissible values of $k$ required above can be weakened
if the equations of state above are considered in the conformal analogues of
the Friedmann and energy conservation equation. In the case of asymptotic dust
the system reads

$(\dot{\Omega})^{2}=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\Omega^{3}\,\frac{\rho}{3},\quad\quad\dot{\rho}=3\,\Omega^{3\,k-1}\,\rho^{1+k}\,w^{*}(\Omega^{3}\,\rho))\,\dot{\Omega},$

in the case of asymptotic radiation the system reads

$(\dot{\Omega})^{2}=\frac{\lambda}{3}-\frac{R[\hat{h}]}{6}\,\Omega^{2}+\Omega^{4}\,\frac{\rho}{3},\quad\quad\dot{\rho}=-3\,\Omega^{4\,k-1}\,\rho^{1+k}\,w^{*}(\Omega^{4}\,\rho))\,\dot{\Omega}.$

For suitable $k$ and initial data these equations can be smoothly integrated
across $\Omega=0$ with $\dot{\Omega}<0$ and $\rho$ bounded and positive. We
shall see now that the FLRW assumption gives quite a wrong impression about
the similarity and simplicity of these two cases if the assumption is dropped.

####  3.3.1 On the equations with an asymptotic dust equation of state.

To illustrate the kind of difficulties which arise in the case of an
asymptotic dust equation of state we consider one of the equations contained
in the complete system of conformal field equations. If the latter are written
in terms of an orthonormal frame $e_{k}$ with $e_{0}=U$, the connection
coefficient $f_{a}=U^{\mu}\,e^{\nu}\,_{a}\,\nabla_{\mu}U_{\nu}$ is subject to
the evolution equation

$e_{0}(f_{a})-w^{\prime}\,e_{a}(\chi_{c}\,^{c})+\chi_{ac}\,f^{c}+(1-3\,w^{\prime})\,L_{a0}-w^{\prime}\,\chi_{c}\,^{c}\,f_{a}$
$=(1-3\,w^{\prime})\,(\Omega^{-1}\,\nabla_{0}\Omega\,f_{a}+\Omega^{-1}\,\chi_{ab}\,\nabla^{b}\Omega-\Omega^{-2}\,\nabla_{0}\Omega\,\nabla_{a}\Omega)$
$-w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}\,(\chi_{c}\,^{c}\,f_{a}-3\,\Omega^{-1}\,\nabla_{0}\,\Omega\,f_{a}-\Omega^{-1}\,\chi_{c}\,^{c}\,\nabla_{a}\Omega+\Omega^{-2}\,\nabla_{0}\Omega\,\nabla_{a}\Omega),$

where $\Omega$, the connection coefficients $\chi_{ab}=\nabla_{a}U_{b}$ and
$\chi_{c}\,^{c}=\nabla_{k}U^{k}$, and the Schouten tensor $L_{ik}$ obey
further evolution equations. In the case of pure dust the third line is not
present. In that case $w^{\prime}=0$ and the second line contains factors
$\Omega^{-1}$. Even if the third line is ignored, the arguments indicated
above in the case of pure dust would not apply in the present case because the
flow equation remains a partial differential equation if $w^{*}>0$,
$\hat{\rho}>0$. It cannot be related to the conformal geodesics equations. The
most complicated term in the equation is, however, the first factor in the
third line. This is not even defined in the case of pure dust. The
$\Omega^{-1}$ terms can not be compensated in this equation by suitable
choices of $k$. We leave this case open.

####  3.3.2 Solutions with an asymptotic radiation equation of state.

In the case of an asymptotic radiation equation of state the situation is
quite different. Consider the equation above again. All the coefficients
introduced by the radiation equation of state are well defined and approach in
the limit $w^{*}\rightarrow 0$ the corresponding value in the case of pure
radiation. Moreover, the terms in the equation above that contain factors
$\Omega^{-1}$ come with the coefficients

$1-3\,w^{\prime}=3\,(\Omega^{4}\,\rho)^{k}\left((k+1)\,w^{*}+\Omega^{4}\,\rho\,(w^{*})^{\prime}\right)\quad\mbox{and}$
$w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}=-(\Omega^{4}\,\rho)^{k}\left(4-3\,\hat{\rho}^{k}\,w^{*}\right)\frac{k\,(1+k)\,w^{*}+2\,(1+k)\,\hat{\rho}\,(w^{*})^{\prime}+\hat{\rho}^{2}\,(w^{*})^{\prime\prime}}{1-3\,(1+k)\,\hat{\rho}^{k}\,w^{*}+3\,\hat{\rho}^{1+k}\,(w^{*})^{\prime}}.$

By a suitable choice of $k$ the factors $\Omega^{4\,k}$ thus allow us to make
up for any negative power of $\Omega$.

The situation is similar with the other equations of the conformal system.
Moreover, the way the asymptotic radiation equation of state affects the
principal part of the system is so that one can still extract from the
complete system in a suitable gauge a reduced system that is symmetric
hyperbolic irrespective of the sign of $\Omega$.

Solutions that admit smooth conformal extensions at future time-like infinity
can now be constructed from data for the conformal field equations which are
given on a space-like hypersurface $S$ in the ‘physical domain’, where
$\Omega>0$ , or from data on the hypersurface $S=\\{\Omega=0\\}$ that
represents future time-like infinity. To check that the data satisfy the
constraints induced on $S$, and possibly the additional requirements implied
by the assumption that $\Omega=0$ on $S$, these conditions are best expressed
in terms of the unit normal to $S$ and then transformed into a frame $e_{k}$
with $e_{0}=U$, whereby the evolution equations need to be used as well.
Unless the field $U$ is assumed to be orthogonal to $S$ this involves some
fairly tedious calculations (see [15], where the presence of a boundary
requires them). Since the latter give limited insight they are skipped here.
After the Cauchy problem has been solved for the given data it follows by
standard arguments that the constraints and thus the complete system of
conformal field equations will be solved as well (see [12], [16] for detailed
discussions). We can state now the following results [14].

The Einstein-$\lambda$-perfect-fluid equations with an asymptotic radiation
equation of state where $k\geq 1$ induce in a suitable gauge a reduced system
of the conformal Einstein-$\lambda$-perfect-fluid equations that is symmetric
hyperbolic irrespective of the sign of $\Omega$.

On a compact 3-dimensional manifold ${\cal J}$ one can construct smooth Cauchy
data for the reduced conformal equations with $\Omega=0$, $U$ time-like future
directed orthogonal to ${\cal J}$, and $<U,d\Omega>\,\,<0$ that satisfy the
constraints induced by the conformal field equations and the special
requirements on a space-like hypersurface on which $\Omega=0$.

These data determine a smooth solution to the reduced equations with $U$
hypersurface orthogonal, $\Omega<0$ in the future of ${\cal J}$ and $\Omega>0$
in the past of ${\cal J}$. In the latter domain the solution defines a unique
solution to the Einstein-$\lambda$-perfect-fluid equations with an asymptotic
radiation equation of state that is time-like geodesically future complete and
for which ${\cal J}$ represents a conformal boundary at the infinite time-like
future.

Let $S$ be a Cauchy hypersurface for this solution in the past of ${\cal J}$
and denote by $\Delta$ the Cauchy data induced by that solution on $S$. Any
Cauchy data $\Delta^{\prime}$ on $S$ for the same equations which are
sufficiently close to $\Delta$ develop into a solution that is also time-like
geodesically future complete, admits a smooth conformal boundary in the
future, and a smooth conformal extension beyond.

We note that the Cauchy hypersurface $S$ above is not required to be
orthogonal to $U$ and that the flow vector field comprised by the data
$\Delta^{\prime}$ on $S$ is not required to satisfy the condition of
hypersurface orthogonality.

###  3.4 Concluding remarks.

If $k$ is large, the terms involving $w^{*}$ in (3.4) and (3.6) may, when
$\Omega\rightarrow 0$, look like minor perturbations of the pure dust or pure
radiation equations of state. We have seen, however, that the effects of these
terms are quite different in the two cases.

When $\Omega$ becomes small the term involving $w^{*}$ may in the case of the
asymptotic radiation equations of state indeed be considered already for $k=1$
as a minor perturbation relative to the dominating first term on the right
hand side of (3.6). This is apparently sufficient to preserve asymptotically
the effects corresponding to the conformal covariance of the pure radiation
equation of state.

In contrast, the transition from the pure to the asymptotic dust equation of
state represented by the term involving $w^{*}$, comes with a drastic change
of the principal part of the matter equations by which the conformal privilege
of the pure dust case is lost completely even if $k$ is large.

One could hope to simplify the analysis of this case by entangling the two
problems that are possibly interfering in the asymptotic dust case at future
time-like infinity. Let the function $w^{*}$ be modified so that
$w^{*}(\hat{\rho})=0$ precisely if $\hat{\rho}=\Omega^{3}\,\rho$ falls below a
certain positive threshold $\hat{\rho}_{*}$. For the sake of discussion assume
that the set $\\{\hat{\rho}=\hat{\rho}_{*}\\}$ defines a space-like Cauchy
hypersurface with the set $\\{\hat{\rho}>\hat{\rho}_{*}\\}$, on which we have
an asymptotic dust equation of state, lying in its past and the set
$\\{\hat{\rho}<\hat{\rho}_{*}\\}$, on which we have a pure dust equation of
state, lying in its future. For this picture to make sense one has to decide
whether the space-time evolution extends with a sufficient degree of
smoothness across the set $\\{\hat{\rho}=\hat{\rho}_{*}\\}$ where the change
of principal part takes place which reduces the PDE for the flow field to an
ODE. Since this concerns the physical domain, one could expect the answer to
follow from the analysis of the fluid equations in [8], [16] which treats the
case $w>0$ and $w=0$ along similar lines. If the answer is positive, the
problem of asymptotic smoothness at time-like infinity only concerns the
fields on $\\{\hat{\rho}<\hat{\rho}_{*}\\}$. This is the situation considered
in [12]. The question of interest now is whether the analyses at the set
$\\{\hat{\rho}=\hat{\rho}_{*}\\}$ and of the behaviour at time-like infinity
can be combined to clarify what happens if the value $\hat{\rho}_{*}$ of the
threshold is lowered to eventually perform the limit
$\hat{\rho}_{*}\rightarrow 0$.

In the light of the preceding discussions I consider this question as
particular interesting. While it may not inform us about the final nature of
the matter fields at the end of our present universe it may give rise to a
more subtle and appropriate definition of an asymptotic dust equation of state
and will certainly give insights into the freedom allowed by the field
equations and the matter equations to model possible ends at future time-like
infinity.

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