#  Cosmological Einstein-$\lambda$-perfect-fluid solutions with asymptotic
dust or radiation equations of state.

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

###### Abstract

This article introduces the notions of asymptotic dust and asymptotic
radiation equations of state. With these non-linear generalizations of the
well known dust or (incoherent) radiation equations of state the perfect-fluid
equations loose any conformal covariance or privilege. We analyse the
conformal field equations induced with these equations of state. It is shown
that the Einstein-$\lambda$-perfect-fluid equations with an asymptotic
radiation equation of state allow for large sets of data that develop into
solutions which admit smooth conformal boundaries in the future and smooth
extensions beyond.

##  1 Introduction

This article extends our investigation of the long term behaviour of solutions
to 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 positive cosmological constant $\lambda$ and energy-momentum tensor
$\hat{T}_{\mu\nu}$ for various matter models. The models considered in the
present paper are given by perfect fluids with certain non-linear equations of
state. Several authors studied the future asymptotic behaviour of solutions to
the Einstein-$\lambda$-perfect fluid equations with a linear equation of state
$\hat{p}=w(\hat{\rho})=w_{*}\,\hat{\rho}$, $w_{*}=const.$ [8], [12], [14],
[15], often assuming that $0<w_{*}<1/3$. More general equations of state have
been considered in the articles [10], [13]. In the present article the
asymptotic behaviour is analysed from a particular point of view which is
motivated by the following observation.

De Sitter space is a geodesically complete, spatially compact, conformally
flat solution of FLRW type to the Einstein-$\lambda$-vacuum equations with
cosmological constant $\lambda>0$. It admits smooth conformal boundaries at
future and past time-like infinity. It generalizes as follows. Smooth Cauchy
data for the same equations on a Cauchy hypersurface $S$ of de Sitter space
that are sufficiently close (with respect to suitable Sobolev norms) to the de
Sitter data develop into solutions that are also time-like and null
geodesically complete and admit smooth conformal boundaries. The conformal
Einstein equations, which have been used for this purpose, determine in fact
smooth solutions that extend beyond these boundaries and define on ‘the other
sides’ again vacuum solutions. Perturbations of the conformal curvature, only
restricted by the smallness condition on $S$, travel unimpeded across the
boundaries [1], [2].

The question of interested in this article is: For which matter models can be
established similar results on the future asymptotic behaviour of solutions to
the Einstein-$\lambda$-matter equations ? For a discussion of the answers
obtained so far the reader is referred to the article [5], which can also be
read as an extended non-technical introduction to the present article. It
further explains our interest in the question above and also may be helpful
for readers not acquainted with the conformal methods used in the following.

The two simplest cases with a positive answer are the FLRW-type solutions to
the Einstein-$\lambda$-perfect-fluid equations with the pure (incoherent)
radiation equation of state $\hat{p}=w(\hat{\rho})=\frac{1}{3}\,\hat{\rho}$
and the pure dust equation of state $\hat{p}=w(\hat{\rho})=0$ (the unusual
word pure has been added here to avoid confusions with notions introduce
below).

These cases have been generalized by showing that these FLRW solutions are
future stable in the class general solutions to the
Einstein-$\lambda$-perfect-fluid equations with the respective equations of
state. The perturbed solutions are not only time-like geodesically future
complete but also admit smooth future conformal boundaries and extensions
beyond [4], [11]. These cases are, however, still somewhat special. The
perfect fluid equations with a pure radiation equation of state considered in
[11] are conformally invariant with vanishing trace of the energy momentum
tensor. The perfect fluid equations with a pure dust equation of state are not
conformally invariant and have an energy momentum tensor with non-vanishing
trace. They are, however, conformally privileged by being related in some
sense to a conformally invariant structure which helps establish the result of
[4].

It may be reasonable to base models of the universe on solutions to the
Einstein-$\lambda$-perfect-fluid equations where the future development is
determined by one of the equations of state above. It appears unlikely,
however, that it makes sense to cover the whole stretch from the Big Bang to
future time-like infinity by one and the same linear equation of state. One
can expect to need a transition where

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

with some function $w^{**}(\hat{\rho})$ of which we will assume that $0\leq
w^{**}(\hat{\rho})\leq 1/3$. At late time $w^{**}(\hat{\rho})$ should then
acquire the value $w^{**}(\hat{\rho})=0$ if we want to model the end by a pure
dust equation of state and $w^{**}(\hat{\rho})=0$ in the case of pure
radiation.

There is nothing, however, which fixes a natural notion of ‘late time’. The
only meaningful requirement would be that these values are assumed in the
limit when the space-time approaches future time-like infinity. But then the
equation of state would still need to recognize where and when this limit will
be achieved.

In the conformal analysis of the two FLRW models mentioned above the physical
density $\hat{\rho}$ and the conformal density $\rho$ satisfy a relation of
the form

$\hat{\rho}=\Omega^{e}\rho,$ (1.2)

with $e=3$ in the case of pure dust and $e=4$ in the case of pure radiation.
These values are chosen because they imply that $\rho=\rho_{*}=const.>0$
(assuming that $\hat{\rho}>0$ on some initial slice). Since then
$\hat{\rho}\rightarrow 0$ at future time-like infinity where
$\Omega\rightarrow 0$, the behaviour of $\hat{\rho}$ can be understood as an
indicator for the approach to future time-like infinity. In generalizing the
situation we shall keep (1.2), hoping it to serve as an indicator function of
the far future where $\Omega\rightarrow 0$. The relation
$\rho=\rho_{*}=const.$ may not be preserved under the generalization
considered below and we will have to control that $\rho$ remains positive and
bounded in the limit $\Omega\rightarrow 0$. The pure dust and pure radiation
equations of state are now generalized as follows.

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

$\hat{p}=w(\hat{\rho})=\left(\hat{\rho}^{k}\,w^{*}(\hat{\rho})\right)\hat{\rho}\quad\mbox{with
some}\quad k\in\mathbb{N},$ (1.3)

combined with (1.2) where $e=3$. It implies
$w^{\prime}(\hat{\rho})=(1+k)\,\hat{\rho}^{k}\,\,w^{*}(\hat{\rho})+\hat{\rho}^{1+k}\,\,(w^{*})^{\prime}(\hat{\rho})$,
where here and in the following the notation
${}^{\prime}=\partial/\partial\hat{\rho}$ is used.

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

$\hat{p}=w(\hat{\rho})=\left(\frac{1}{3}-\hat{\rho}^{k}\,\,w^{*}(\hat{\rho})\right)\hat{\rho}\quad\mbox{with
some}\quad k\in\mathbb{N},$ (1.4)

combined with (1.2) where $e=4$. It implies
$w^{\prime}(\hat{\rho})=\frac{1}{3}-(1+k)\,\hat{\rho}^{k}\,\,w^{*}(\hat{\rho})-\hat{\rho}^{1+k}\,\,(w^{*})^{\prime}(\hat{\rho})$.
In both case $w^{*}(\hat{\rho})$ is a smooth functionof $\hat{\rho}$ that
satisfies

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

To ensure that $0\leq w^{**}(\hat{\rho})\leq 1/3$ as required above, we would
need to impose more detailed conditions on $w^{*}(\hat{\rho})$ but for the
analysis of the effect of the new equations of state in the far future where
$\Omega$ becomes small the conditions above are sufficient. The limits
$w^{*}(\hat{\rho})\rightarrow 0$ give back the pure dust and the pure
radiation equations of state.

The factor $\hat{\rho}^{k}$ has been inserted in the definitions as a simple
means to control in terms of $k$ the speed at which the pure dust or the pure
radiation situations is approximated as $\hat{\rho}\rightarrow 0$. Eventually
the conformal field equations may force us to impose less crude conditions on
$w^{*}(\hat{\rho})$ or to introduce more subtle definitions of asymptotic
equations of state.

Definition (1.3) implies

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

while definition (1.4) 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.

Because the principal parts of the matter equations are affected by the
equations of state above any conformal covariance or privilege is lost.

The Cauchy problem local in time for Einstein-$\lambda$-perfect fluids with an
asymptotic dust or radiation equations of state as above poses no problems.
This follows from the results of [3], [7] where only weak conditions on the
equation of state are assumed.

First conditions on the admissible values of $k$ are found if the equations of
state above are considered in the conformal analogues of the FLRW Friedmann
and the energy conservation equation. In the case of asymptotic dust the
systems reads

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

where $R[\hat{h}]=const.\geq 0$. It can be integrated across $\Omega=0$ with
$\rho$ bounded and positive if $3\,k-1\in\mathbb{N}_{0}$. In the case of
asymptotic radiation the systems reads

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

It can be integrated across $\Omega=0$ with $\rho$ bounded and positive if
$4\,k-1\in\mathbb{N}_{0}$.

In the following sections we derive the conformal Einstein-$\lambda$-perfect
fluid equations, introduce a gauge that involves in particular an orthonormal
frame $e_{j}$, $j=0,1,2,3$, with $e_{0}=U$, and discuss the regularity as
$\Omega\rightarrow 0$ of the equations in this gauge with any of the two
asymptotic equations of state assumed.

In the case of an asymptotic dust equations of state there arise problems. The
use of the asymptotic equation of state affects the principal part of the
fluid equations in a way which does not allow us to apply the methods which
were successful in the case of the pure dust equation of state [4]. This case,
which I consider as particularly interesting (see [5]), is left open.

In the case of an asymptotic radiation equation of state regularity of the
equations can be ensured by a suitable assumption and there can be derived a
symmetric hyperbolic reduced system that is well defined irrespective of the
sign of $\Omega$. This allows us to give the following answer to the question
posed in the beginning.

Consider the reduced system of the conformal Einstein-$\lambda$-perfect-fluid
equations in the gauge discussed below with an asymptotic radiation equation
of state where $k\geq 1$. On a compact 3-dimensional manifold ${\cal J}^{+}$
let be given smooth Cauchy data for the reduced equations with $\Omega=0$,
$U^{k}=\delta^{k}\,_{0}$ time-like future directed, and
$U^{k}\,\nabla_{k}\Omega<0$ that satisfy the constraints induced by the
conformal field equations and the special properties implied on a space-like
hypersurface on which $\Omega=0$. These data determine a smooth solution to
the reduced equations with $\Omega<0$ in the future of ${\cal J}^{+}$ and
$\Omega>0$ in the past of ${\cal J}^{+}$. Where $\Omega>0$ the solution
defines a unique solution to the system of 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 the
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 the solution on $S$.
Any Cauchy data $\Delta^{\prime}$ on $S$ for the same system which are
sufficiently close to $\Delta$ develop into a solution which is also time-like
geodesically future complete, admits a smooth conformal boundary in its
future, and a smooth conformal extension beyond.

At the end of this article are discussed observations about solutions for
which $U$ is orthogonal to the hypersurface ${\cal J}^{+}$ and consequently
hypersurface orthogonal on physical solution in the past of ${\cal J}^{+}$.

##  2 The conformal Einstein-$\lambda$-Euler system.

We consider the Einstein-$\lambda$-perfect fluid system with cosmological
constant $\lambda$ that consists of the Einstein equations (1.1) with
cosmological constant $\lambda>0$ and an energy momentum tensor of a simple
ideal fluid given by

$\hat{T}_{\mu\nu}=(\hat{\rho}+\hat{p})\,\hat{U}_{\mu}\,\hat{U}_{\nu}+\hat{p}\,\hat{g}_{\mu\nu}.$

The unknowns are a Lorentz metric $\hat{g}_{\mu\nu}$ on a four-dimensional
manifold $\hat{M}$, a future directed time-like fluid flow vector field
$\hat{U}^{\mu}$ satisfying $\hat{U}_{\mu}\,\hat{U}^{\mu}=-1$, the total energy
density $\hat{\rho}$ and the pressure $\hat{p}$ as measured by co-moving
observers. The system is be completed by an equation of state

$\hat{p}=w(\hat{\rho})\geq 0.$ (2.1)

The matter equations, equivalent to $\hat{\nabla}^{\mu}\,\hat{T}_{\mu\nu}=0$,
are given by

$\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\,\hat{\rho}+(\hat{\rho}+\hat{p})\,\hat{\nabla}_{\mu}\,\hat{U}^{\mu}=0,$
(2.2)
$(\hat{\rho}+\hat{p})\,\hat{U}^{\mu}\,\hat{\nabla}_{\mu}\,\hat{U}_{\nu}+\\{\hat{U}^{\mu}\,\hat{U}_{\nu}+\hat{g}^{\mu}\,_{\nu}\\}\,\hat{\nabla}_{\mu}\hat{p}=0.$
(2.3)

We assume that $\hat{\rho}>0$ on an initial Cauchy slice. The first of the
matter equations then ensures that this relation is preserved where the
solution to the equations is regular. Let $\hat{e}_{a}$, $a=1,2,3$, be vector
fields that satisfy $\hat{g}(\hat{U},\hat{e}_{a})=0$,
$\hat{g}(\hat{e}_{a},\hat{e}_{b})=0$ and $F=F(\hat{\rho})$ a function that
satisfies

$F^{\prime}=-(\hat{\rho}+w)^{-1}\,w^{\prime},$ (2.4)

where we use, as in the following chapters, the notation
${}^{\prime}=\frac{\partial}{\partial\hat{\rho}}$. With (2.3) follows

$\hat{f}_{a}\equiv\hat{U}^{\mu}\,\hat{e}^{\nu}\,_{a}\,\hat{\nabla}_{\mu}\,\hat{U}_{\nu}=\hat{e}_{a}(F).$
(2.5)

Equations (2.2), (2.3) imply

$w^{\prime}\,\hat{\nabla}_{\nu}\hat{\rho}=(\hat{\rho}+w)\,\left(w^{\prime}\,\hat{U}_{\nu}\,\hat{\nabla}_{\rho}\,\hat{U}^{\rho}-\hat{U}^{\rho}\,\hat{\nabla}_{\rho}\,\hat{U}_{\nu}\right).$
(2.6)

Taking a derivative and observing that $\hat{\rho}+w>0$ we get the equation

$w^{\prime\prime}\,\hat{\nabla}_{\mu}\hat{\rho}\,\,\hat{\nabla}_{\nu}\hat{\rho}+w^{\prime}\,\,\hat{\nabla}_{\mu}\,\hat{\nabla}_{\nu}\hat{\rho}=w^{\prime}\,\frac{1+w^{\prime}}{\hat{\rho}+w}\,\hat{\nabla}_{\mu}\hat{\rho}\,\hat{\nabla}_{\nu}\hat{\rho}$
$+(\hat{\rho}+w)\left(w^{\prime\prime}\,\hat{\nabla}_{\mu}\hat{\rho}\,\,\hat{U}_{\nu}\,\hat{\nabla}_{\rho}\,\hat{U}^{\rho}+w^{\prime}\,\hat{\nabla}_{\mu}\,\hat{U}_{\nu}\,\hat{\nabla}_{\rho}\,\hat{U}^{\rho}+w^{\prime}\,\hat{U}_{\nu}\,\hat{\nabla}_{\mu}\,\hat{\nabla}_{\rho}\,\hat{U}^{\rho}\right)$
$-(\hat{\rho}+w)\left(\hat{\nabla}_{\mu}\hat{U}^{\rho}\,\hat{\nabla}_{\rho}\,\hat{U}_{\nu}+\hat{U}^{\rho}\,\hat{\nabla}_{\mu}\hat{\nabla}_{\rho}\,\hat{U}_{\nu}\right).$

Where $w^{\prime}\neq 0$ anti-symmetrization of this relation implies the
integrability condition

$\hat{U}^{\rho}\,\hat{\nabla}_{\rho}\hat{\nabla}_{[\mu}\,\hat{U}_{\nu]}-\hat{\nabla}_{\rho}\,\hat{U}_{[\mu}\,\hat{\nabla}_{\nu]}\hat{U}^{\rho}=$
(2.7)
$w^{\prime}\,\hat{\nabla}_{[\mu}\,\hat{U}_{\nu]}\,\hat{\nabla}_{\rho}\,\hat{U}^{\rho}-w^{\prime}\,\hat{U}_{[\mu}\,\hat{\nabla}_{\nu]}\,\hat{\nabla}_{\rho}\,\hat{U}^{\rho}-w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}\,\hat{\nabla}_{\lambda}\,\hat{U}^{\lambda}\,\hat{U}^{\rho}\,\hat{\nabla}_{\rho}\,\hat{U}_{[\mu}\,\hat{U}_{\nu]}.$

Conformal transformations of the curvature fields.

The conformal Weyl tensor $C^{\mu}\,_{\nu\lambda\eta}$ and the Schouten tensor
$L_{\mu\nu}=1/2\left(R_{\mu\nu}-1/6\,R\,g_{\mu\nu}\right)$ of a given metric
satisfy the contracted Bianchi identity

$\nabla_{\mu}\,C^{\mu}\,_{\nu\lambda\eta}=2\,\nabla_{[\lambda}\,L_{\eta]\nu}.$

Under the rescaling

$\hat{g}_{\mu\nu}\rightarrow
g_{\mu\nu}=\Omega^{2}\,\hat{g}_{\mu\nu},\quad\Omega>0,$

the conformal Weyl tensor is invariant,
$C^{\mu}\,_{\nu\lambda\eta}=\hat{C}^{\mu}\,_{\nu\lambda\eta}$, the Ricci
tensor transform as

$\hat{R}_{\mu\nu}\rightarrow
R_{\mu\nu}=\hat{R}_{\mu\nu}-2\,\Omega^{-1}\nabla_{\mu}\nabla_{\nu}\Omega-g^{\lambda\delta}\left(\Omega^{-1}\,\nabla_{\lambda}\nabla_{\delta}\Omega-3\,\Omega^{-2}\nabla_{\lambda}\Omega\nabla_{\delta}\Omega\right)g_{\mu\nu},$

and the Schouten tensor as

$\hat{L}_{\mu\nu}\rightarrow
L_{\mu\nu}=\hat{L}_{\mu\nu}-\Omega^{-1}\nabla_{\mu}\nabla_{\nu}\Omega+1/2\,\,\Omega^{-2}\,g^{\lambda\delta}\,\nabla_{\lambda}\Omega\,\nabla_{\delta}\Omega\,g_{\mu\nu}.$
(2.8)

The conformal Weyl tensor satisfies
$\nabla_{\mu}\left(\Omega^{-1}\,C^{\mu}\,_{\nu\lambda\eta}\right)=\Omega^{-1}\,\hat{\nabla}_{\mu}\,\hat{C}^{\mu}\,_{\nu\lambda\eta}$
where $\nabla$ and $\hat{\nabla}$ denote the Levi-Civita connections of
$g_{\mu\nu}$ and $\hat{g}_{\mu\nu}$. The identity above thus implies for the
rescaled conformal Weyl tensor
$W^{\mu}\,_{\nu\lambda\eta}=\Omega^{-1}\,C^{\mu}\,_{\nu\lambda\eta}$ the
relation

$\nabla_{\mu}\,W^{\mu}\,_{\nu\lambda\eta}=2\,\Omega^{-1}\,\hat{\nabla}_{[\lambda}\,\hat{L}_{\eta]\nu},$
(2.9)

whence

$2\,\nabla_{[\lambda}\,L_{\eta]\nu}-\nabla_{\mu}\,\Omega\,W^{\mu}\,_{\nu\lambda\eta}=2\,\hat{\nabla}_{[\lambda}\,\hat{L}_{\eta]\nu}.$
(2.10)

Conformal transformation of the matter fields.

We combine the conformal rescaling of the metric with the rescalings

$U^{\mu}=\Omega^{-1}\,\hat{U}^{\mu}\quad
U_{\mu}=\Omega\,\hat{U}_{\mu},\quad\rho=\Omega^{-e}\,\hat{\rho},\quad\mbox{with
$e=const.>0$.}$

In the following it will always be understood that

$w=w(\hat{\rho})=w(\Omega^{e}\,\rho),\quad\quad
w^{\prime}=w^{\prime}(\hat{\rho})=w^{\prime}(\Omega^{e}\,\rho),$

and similarly with the function $w^{*}$ introduced later. Equations (1.1)
imply

$\hat{T}=\hat{g}^{\mu\nu}\,\hat{T}_{\mu\nu}=3\,w-\hat{\rho},\quad\quad\hat{R}=4\,\lambda-\hat{T}=4\,\lambda+\hat{\rho}-3\,w,$

and
$\hat{L}_{\mu\nu}=\hat{L}^{*}_{\mu\nu}+\frac{1}{6}\,\lambda\,\hat{g}_{\mu\nu}$
with

$\hat{L}^{*}_{\mu\nu}=\frac{1}{2}\,(\hat{\rho}+w)\,\hat{U}_{\mu}\,\hat{U}_{\nu}+\frac{1}{6}\,\hat{\rho}\,\hat{g}_{\mu\nu}=\Omega^{-2}\left(\frac{1}{2}\,(\hat{\rho}+w)\,U_{\mu}\,U_{\nu}+\frac{1}{6}\,\hat{\rho}\,g_{\mu\nu}\right),$
(2.11)

It follows that
$\hat{\nabla}_{\lambda}\hat{L}_{\eta\nu}=\hat{\nabla}_{\lambda}\hat{L}^{*}_{\eta\nu}$.
The Ricci scalar of the conformal metric satisfies

$6\,\Omega\,\nabla_{\mu}\nabla^{\mu}\Omega+\Omega^{2}\,R-12\,\nabla_{\mu}\Omega\nabla^{\mu}\Omega=4\,\lambda+\hat{\rho}-3\,w.$

Written in the form

$6\,\Omega\,s-3\,\nabla_{\mu}\Omega\nabla^{\mu}\Omega=\lambda+\frac{1}{4}\,(\hat{\rho}-3\,w)\quad\mbox{with}\quad
s=\frac{1}{4}\,\nabla_{\mu}\nabla^{\mu}\Omega+\frac{1}{24}\,\Omega\,R,$ (2.12)

it will be referred to as $\lambda$-equation. Equations (2.8), (2.11), (2.12)
give

$\nabla_{\mu}\nabla_{\nu}\Omega=-\Omega\,L_{\mu\nu}+s\,g_{\mu\nu}+\Omega^{-1}\,\frac{\hat{\rho}+w}{2}\left(U_{\mu}\,U_{\nu}+\frac{1}{4}\,g_{\mu\nu}\right)$
(2.13)

which will be referred to as $\Omega$-equation. Applying a derivative to the
$\lambda$-equation (2.12) gives with (2.13) the $s$-equation

$\nabla_{\mu}s+L_{\mu\nu}\,\nabla^{\nu}\Omega=\Omega^{-2}\,\frac{\hat{\rho}+w}{2}\left(U_{\mu}\,U^{\nu}\,\nabla_{\nu}\Omega+\frac{1}{4}\,\nabla_{\mu}\Omega\right)+\frac{1}{24}\,(1-3\,w^{\prime})\Omega^{-1}\,\nabla_{\mu}\hat{\rho}.$
(2.14)

It follows

$\hat{\nabla}_{[\lambda}\,\hat{L}_{\eta]\nu}=\nabla_{[\lambda}\,\hat{L}^{*}_{\eta]\nu}-\Omega^{-1}\left(\hat{L}^{*}_{\nu[\lambda}\,\nabla_{\eta]}\,\Omega+g_{\nu[\lambda}\,\hat{L}^{*}_{\eta]\rho}\,\nabla^{\rho}\Omega\right),$

which gives with (2.11) and
$\nabla_{\lambda}\hat{\rho}=\Omega^{e}\,(\nabla_{\lambda}\rho+e\,\rho\,\Omega^{-1}\,\nabla_{\lambda}\Omega)$
the relation

$2\,\hat{\nabla}_{[\lambda}\,\hat{L}_{\eta]\nu}=\Omega^{-3}\,(\hat{\rho}+w)\left(U_{\nu}\,U_{[\lambda}\,\nabla_{\eta]}\,\Omega-
g_{\nu[\lambda}\,U_{\eta]}\,U_{\rho}\,\nabla^{\rho}\Omega\right)$ (2.15)
$+e\,\rho\,\,\Omega^{e-3}\left((1+w^{\prime})\,\nabla_{[\lambda}\Omega\,U_{\eta]}\,U_{\nu}+\frac{1}{3}\,\nabla_{[\lambda}\Omega\,g_{\eta]\nu}\right)$
$+\Omega^{-2}\,(\hat{\rho}+w)\left(\nabla_{[\lambda}\,U_{\eta]}\,U_{\nu}+U_{[\eta}\,\nabla_{\lambda]}\,U_{\nu}\right)$
$+\Omega^{e-2}\,\left((1+w^{\prime})\,\nabla_{[\lambda}\,\rho\,\,U_{\eta]}\,U_{\nu}+\frac{1}{3}\,\nabla_{[\lambda}\,\rho\,g_{\eta]\nu}\right).$

The conformal matter equations

Equations (2.2) and (2.3) transform with
$\nabla_{\mu}\hat{\rho}=\Omega^{e}\left\\{\nabla_{\mu}\,\rho+e\,\rho\,\Omega^{-1}\,\nabla_{\mu}\Omega\right\\}$
into

$0=U^{\mu}\,\nabla_{\mu}\rho+(\rho+\Omega^{-e}\,w)\,\nabla_{\mu}\,U^{\mu}-\Omega^{-1}\,(-e\,\rho+3\,\rho+3\,\Omega^{-e}\,w)\,U^{\mu}\,\nabla_{\mu}\Omega,$
(2.16)
$0=(\rho+\Omega^{-e}\,w)\,U^{\mu}\,\nabla_{\mu}\,U_{\nu}+w^{\prime}\,(g^{\mu}\,_{\nu}+U^{\mu}\,U_{\nu})\,\nabla_{\mu}\,\rho\quad\quad\quad\quad\quad\quad\quad\,\,\,$
(2.17)
$\quad\quad\quad\quad\quad-\Omega^{-1}\,(\rho+\Omega^{-e}\,w-e\,\rho\,w^{\prime})(g^{\mu}\,_{\nu}+U^{\mu}\,U_{\nu})\nabla_{\mu}\Omega.$

In the case of pure dust, where $w=0$ and $e=3$, the equations read

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

The $\Omega^{-1}$ term in the second equation reflects the conformal non-
covariance of the system. In the case of pure radiation with
$w=\frac{1}{3}\,\hat{\rho}$, and $e=4$ the conformal equations reduce to

$U^{\mu}\,\nabla_{\mu}\rho+\frac{4}{3}\,\rho\,\nabla_{\mu}\,U^{\mu}=0,\quad\quad\frac{4}{3}\,\rho\,\,U^{\mu}\,\nabla_{\mu}\,U_{\nu}+\frac{1}{3}\,(g^{\mu}\,_{\nu}+U^{\mu}\,U_{\nu})\,\nabla_{\mu}\,\rho=0,$

and have thus the same form as their physical versions.

Equation (2.7) transforms into

$U^{\rho}\,\nabla_{\rho}\nabla_{[\mu}\,U_{\nu]}+w^{\prime}\,U_{[\mu}\,\nabla_{\nu]}\,\nabla_{\rho}\,U^{\rho}$
(2.18)
$-\nabla_{\pi}\,U_{[\mu}\,\nabla_{\nu]}\,U^{\pi}-(1-3\,w^{\prime})\,U_{[\mu}\,L_{\nu]\rho}\,U^{\rho}-w^{\prime}\,\nabla_{[\mu}\,U_{\nu]}\,\nabla_{\pi}\,U^{\pi}=$
$+(1-3\,w^{\prime})\left\\{\Omega^{-1}\,\nabla_{[\mu}\,U_{\nu]}\,U^{\rho}\,\nabla_{\rho}\Omega-\Omega^{-1}\,U_{[\mu}\,\nabla_{\nu]}\,U_{\rho}\,\nabla^{\rho}\Omega+\Omega^{-2}\,U_{[\mu}\,\nabla_{\nu]}\Omega\,U^{\rho}\,\nabla_{\rho}\Omega\right\\}$
$-w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}\,\left\\{\nabla_{\pi}\,U^{\pi}\,U^{\rho}\,\nabla_{\rho}\,U_{[\mu}\,U_{\nu]}-3\,\Omega^{-1}\,U^{\pi}\,\nabla_{\pi}\Omega\,U^{\rho}\,\nabla_{\rho}\,U_{[\mu}\,U_{\nu]}\right.$
$\left.-\Omega^{-1}\,\nabla_{\pi}\,U^{\pi}\,\nabla_{[\mu}\Omega\,U_{\nu]}+3\,\Omega^{-2}\,U^{\pi}\,\nabla_{\pi}\Omega\,\nabla_{[\mu}\Omega\,U_{\nu]}\right\\}.$

With $\hat{e}^{\mu}\,_{a}=\Omega\,e^{\mu}\,_{a}$ the relation (2.5) transforms
into

$e_{a}(F)=\Omega^{-1}\,\hat{f}_{a}=U^{\mu}\,\nabla_{\mu}U_{\nu}\,e^{\nu}\,_{a}-e_{a}(\log\Omega).$
(2.19)

##  3 The gauge and the implied equations.

We express the equations in terms of an orthonormal frame field
$e_{k}=e^{\mu}\,_{k}\partial_{x^{\mu}}$, $k=0,1,2,3$, so that $g_{jk}\equiv
g(e_{j},e_{k})=\eta_{jk}=diag(-1,1,1,1)$ and $e_{0}$ is a time-like vector
field. The space-like frame vector fields are then given by the $e_{a}$, where
$a,b,c=1,2,3$ denote spatial indices to which the summation convention also
applies. In the following all tensor fields considered in the previous
sections will be expressed in terms of this frame field. The contravariant
coordinate version of the metric is given by
$g^{\mu\nu}=\eta^{jk}\,e^{\mu}\,_{j}\,e^{\nu}\,_{k}$.

The connection coefficients, defined by
$\nabla_{j}e_{k}\equiv\nabla_{e_{j}}e_{k}=\Gamma_{j}\,^{l}\,{}_{k}\,e_{l}$,
satisfy $\Gamma_{jlk}=-\Gamma_{jkl}$ with
$\Gamma_{jlk}=\Gamma_{j}\,^{i}\,{}_{k}\,g_{li}$, because $\nabla_{i}g_{jk}=0$.
The covariant derivative of a tensor field $X^{\mu}\,_{\nu}$, given in the
frame by $X^{i}\,_{j}$, takes the form
$\nabla_{k}\,X^{i}\,_{j}=X^{i}\,_{j\,,\mu}\,e^{\mu}\,_{k}+\Gamma_{k}\,^{i}\,{}_{l}\,X^{l}\,_{j}-\Gamma_{k}\,^{i}\,{}_{l}\,X^{i}\,_{j}$.
The frame and the connection coefficients satisfy the first structural
equations

$e^{\mu}\,_{i,\,\nu}\,e^{\nu}\,_{j}-e^{\mu}\,_{j,\,\nu}\,e^{\nu}\,_{i}=(\Gamma_{j}\,^{k}\,{}_{i}-\Gamma_{i}\,^{k}\,{}_{j})\,e^{\mu}\,_{k},$
(3.1)

which ensures that the connection is torsion free, and the second structural
equations

$\Gamma_{l}\,^{i}\,{}_{j,\,\mu}\,e^{\mu}\,_{k}-\Gamma_{k}\,^{i}\,{}_{j,\,\mu}\,e^{\mu}\,_{l}+2\,\Gamma_{[k}\,^{i\,p}\,\Gamma_{l]pj}-2\,\Gamma_{[k}\,^{p}\,{}_{l]}\,\Gamma_{p}\,^{i}\,{}_{j}$
(3.2)
$=\Omega\,W^{i}\,_{jkl}+2\,\\{g^{i}\,_{[k}\,L_{l]j}+L^{i}\,_{[k}\,g_{l]j}\\}.$

To restrict the gauge freedom for the frame we set $e_{0}=U$ so that
$U=U^{k}\,e_{k}$ with $U^{k}=\delta^{k}\,_{0}$, choose at the points of a
given smooth space-like hypersurface $S$ transverse to the flow line of $U$
vector fields $e_{a}$, $a=1,2,3$, so that $g(e_{j},e_{k})=\eta_{jk}$, assume
the orthonormal frame $e_{k}$ to be extended by Fermi-transport in the
direction of $e_{0}=U$ so that
$0=\mathbb{F}_{U}e_{k}=\nabla_{U}e_{k}-g(e_{k},\nabla_{U}U)\,U+g(e_{k},U)\,\nabla_{U}U$,
and assume the $e_{a}$ to be chosen on $S$ so that the resulting orthonormal
frame field is smooth. In terms of this frame the Fermi transport law reduces
to

$\Gamma_{0}\,^{a}\,{}_{b}=0.$ (3.3)

To restrict the gauge freedom for the coordinates $x^{\mu}$, we assume that
$\tau\equiv x^{0}=0$ on $S$ and the $x^{\mu}$ be dragged along with $U$ so
that

$<e_{0},dx^{\mu}>\,=e^{\mu}\,_{0}=U^{\mu}=\delta^{\mu}\,_{0}.$

The remaining non-vanishing connection coefficients are given by

$f_{a}\equiv\Gamma_{0}\,^{0}\,{}_{a}=\delta_{ab}\,\,\Gamma_{0}\,^{b}\,{}_{0},\quad\quad\chi_{ab}\equiv\Gamma_{a}\,^{0}\,{}_{b}=\delta_{bc}\,\,\Gamma_{a}\,^{c}\,{}_{0}\quad\mbox{and}\quad\Gamma_{a}\,^{b}\,{}_{c}.$

It holds
$\nabla_{i}\,U_{k}=\Gamma_{i}\,^{0}\,{}_{k}=\chi_{ab}\,\delta^{a}\,_{i}\,\delta^{b}\,_{k}+f_{b}\,\delta^{0}\,_{i}\,\delta^{b}\,_{k}$
and $U^{i}\,\nabla_{i}\,U_{k}=f_{b}\,\delta^{b}\,_{k}$,
$\nabla_{i}\,U^{i}=\chi_{a}\,^{a}$.

The first structural equations supply the constraint

$e^{\mu}\,_{a,\,\nu}\,e^{\nu}\,_{b}-e^{\mu}\,_{b,\,\nu}\,e^{\nu}\,_{a}=(\Gamma_{b}\,^{c}\,{}_{a}-\Gamma_{a}\,^{c}\,{}_{b})\,e^{\mu}\,_{c}+(\chi_{ba}-\chi_{ab})\,\delta^{\mu}\,_{0},$
(3.4)

and the evolution equations

$e^{\mu}\,_{a,\,0}=-\chi_{a}\,^{b}\,e^{\mu}\,_{b}+f_{a}\,\delta^{\mu}\,_{0}.$
(3.5)

The second structural equations imply for $\Gamma_{a}\,^{b}\,{}_{c}$ and
$\chi_{ab}$ the constraints

$\Gamma_{b}\,^{a}\,{}_{c,\,\mu}\,e^{\mu}\,_{d}-\Gamma_{d}\,^{a}\,{}_{c,\,\mu}\,e^{\mu}\,_{b}+2\,\Gamma_{[d}\,^{a\,i}\,\Gamma_{b]ic}-2\,\Gamma_{[d}\,^{i}\,{}_{b]}\,\Gamma_{i}\,^{a}\,{}_{c}$
(3.6)
$=\Omega\,W^{a}\,_{cdb}+2\,\\{g^{a}\,_{[d}\,L_{b]c}+L^{a}\,_{[d}\,g_{b]c}\\},$
$\chi_{ab,\,\mu}\,e^{\mu}\,_{c}-\chi_{cb,\,\mu}\,e^{\mu}\,_{a}+2\,\Gamma_{[c}\,^{0\,p}\,\Gamma_{a]pb}-2\,\Gamma_{[c}\,^{p}\,{}_{a]}\,\Gamma_{p}\,^{0}\,{}_{b}=\Omega\,W^{0}\,_{bca}+2\,L^{0}\,_{[c}\,g_{a]b},$
(3.7)

and the evolution equations

$\Gamma_{a}\,^{b}\,{}_{c,\,0}+f^{b}\,\chi_{ac}-\chi_{a}\,^{b}\,f_{c}+\chi_{a}\,^{d}\,\Gamma_{d}\,^{b}\,{}_{c}=\Omega\,W^{b}\,_{c0a}-g^{b}\,_{a}\,L_{0c}+L^{b}\,_{0}\,g_{ac},$
(3.8)
$\chi_{ac,\,0}-f_{c,\,\mu}\,e^{\mu}\,_{a}+f_{b}\,\Gamma_{a}\,^{b}\,{}_{c}-f_{a}\,f_{b}+\chi_{a}\,^{b}\,\chi_{bc}=\Omega\,W^{0}\,_{c0a}+L_{ac}+L^{0}\,_{0}\,g_{ac}.$
(3.9)

No equation for $f_{a}$ is implied by the structural equations.

The fluid equations (2.16), (2.17) imply in our gauge the evolution equation

$\nabla_{0}\rho+(\rho+\Omega^{-e}\,w)\,\chi_{a}\,^{a}-\Omega^{-1}\,(-e\,\rho+3\,\rho+3\,\Omega^{-e}\,w)\,\nabla_{0}\Omega=0,$
(3.10)

and the constraint

$(\rho+\Omega^{-e}\,w)\,f_{a}+w^{\prime}\,\nabla_{a}\,\rho-\Omega^{-1}\,(\rho+\Omega^{-e}\,w-e\,\rho\,w^{\prime})\nabla_{a}\Omega=0.$
(3.11)

With $\hat{\rho}=\Omega^{e}\,\rho$ the latter is seen to be (2.19), which
reads with $f_{a}=U^{\mu}\,\nabla_{\mu}U_{\nu}\,e^{\nu}\,_{a}$

$f_{a}=e_{a}(F+\log\Omega).$ (3.12)

Equation (2.18) is equivalent to the two equations

$e_{0}(f_{a})-w^{\prime}\,e_{a}(\chi_{c}\,^{c})+\chi_{ac}\,f^{c}+(1-3\,w^{\prime})\,L_{a0}-w^{\prime}\,\chi_{c}\,^{c}\,f_{a}=$
(3.13)
$=(1-3\,w^{\prime})\,(\Omega^{-1}\,\nabla_{0}\Omega\,f_{a}+\Omega^{-1}\,\chi_{ab}\,\nabla^{b}\Omega-\Omega^{-2}\,\nabla_{0}\Omega\,\nabla_{a}\Omega)$
$-w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}\,(\chi_{c}\,^{c}\,f_{a}-3\,\Omega^{-1}\,\nabla_{0}\,\Omega\,f_{a}-\Omega^{-1}\,\chi_{c}\,^{c}\,\nabla_{a}\Omega+\Omega^{-2}\,\nabla_{0}\Omega\,\nabla_{a}\Omega),$

and

$e_{0}(\chi_{[ab]})-\chi_{c[a}\,\chi_{b]}\,^{c}-w^{\prime}\,\chi_{c}\,^{c}\,\chi_{[ab]}=(1-3\,w^{\prime})\,\Omega^{-1}\,\nabla_{0}\Omega\,\chi_{[ab]}.$
(3.14)

With
$\chi_{c[a}\,\chi_{b]}\,^{c}=-\chi_{[ac]}\,\sigma_{b}\,^{c}+\chi_{[bc]}\,\sigma_{a}\,^{c}$,
where $\sigma_{ab}=\chi_{(ab)}$, the latter can be read as a linear
homogeneous ODE for $\chi_{[ab]}$.

The system of conformal field equations reads with $e=const.$

$6\,\Omega\,s-3\,\nabla_{k}\Omega\,\nabla^{k}\Omega-\lambda=\frac{1}{4}\,(\hat{\rho}-3\,w),$
(3.15)

$\nabla_{k}\,\nabla_{j}\Omega+\,\Omega\,L_{kj}-s\,g_{kj}=\Omega^{-1}\,\frac{\hat{\rho}+w}{2}\left(U_{k}\,U_{j}+\frac{1}{4}\,g_{kj}\right),$
(3.16)

$\nabla_{k}\,s+\nabla^{i}\Omega\,L_{ik}=$ (3.17)
$\Omega^{-2}\,\frac{\hat{\rho}+w}{2}\left(U_{k}\,U^{i}\,\nabla_{i}\Omega+\frac{1}{4}\,\nabla_{k}\Omega\right)+\frac{1}{24}\,(1-3\,w^{\prime})\,\Omega^{e-1}\left\\{\nabla_{\mu}\,\rho+e\,\rho\,\Omega^{-1}\,\nabla_{k}\Omega\right\\},$

$\nabla_{i}\,L_{jk}-\nabla_{j}\,L_{ik}-\nabla_{l}\Omega\,\,W^{l}\,_{kij}=2\,\hat{\nabla}_{[i}\,\hat{L}_{j]k},$
(3.18)

$\nabla_{l}\,W^{l}\,_{kij}=2\,\Omega^{-1}\,\hat{\nabla}_{[i}\,\hat{L}_{j]k},$
(3.19)

with

$2\,\hat{\nabla}_{[i}\,\hat{L}_{j]k}=\Omega^{-3}\,(\hat{\rho}+w)\left(U_{k}\,U_{[i}\,\nabla_{j]}\,\Omega-
g_{k[i}\,U_{j]}\,U^{l}\,\nabla_{l}\Omega\right)$ (3.20)
$+e\,\rho\,\Omega^{e-3}\left((1+w^{\prime})\,\nabla_{[i}\Omega\,U_{j]}\,U_{k}+\frac{1}{3}\,\nabla_{[i}\Omega\,g_{j]k}\right)$
$+\Omega^{-2}\,(\hat{\rho}+w)\left(\nabla_{[i}\,U_{j]}\,U_{k}+U_{[j}\,\nabla_{i]}\,U_{k}\right)$
$+\Omega^{e-2}\,\left((1+w^{\prime})\,\nabla_{[i}\,\rho\,\,U_{j]}\,U_{k}+\frac{1}{3}\,\nabla_{[i}\,\rho\,g_{j]k}\right).$

##  4 Regularity of the equations.

In the following an equation will be called ‘regular’ if no negative or non-
integer powers of $\Omega$ occur in it. The structural equations are regular
in this sense but the tensorial equations can be problematic. It will use from
now on the notation

$\nabla_{k}\Omega=\Sigma_{k}.$ (4.1)

The asymptotic dust equation of state.

With the asymptotic dust equation of state and $e=3$ there enter in various
places of the equations functions such as

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

As $w^{*}\rightarrow 0$ they approach the values of the corresponding
quantities of the pure dust equation of state. Therefore it is clear that
equations which are singular in that case must also be singular in the present
case.

In the present case the singular terms cannot be handled as in the pure dust
case because equation (2.3) remains a partial differential equation and the
$\Omega^{-1}$ terms cannot be compensated by suitable choices of $k$.
Moreover, the expression

$w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}=(1+\hat{\rho}^{k}\,w^{*})\left(k+\hat{\rho}\,\frac{(2+k)\,(w^{*})^{\prime}+\hat{\rho}\,(w^{*})^{\prime\prime}}{(1+k)\,w^{*}+\hat{\rho}\,(w^{*})^{\prime}}\right),$

which comes with a singular factor in the equation

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

is not even defined if $w^{\prime}=0$. Since there is no obvious way to handle
it, this case will not be considered any further in this article.

The asymptotic radiation equation of state.

The asymptotic radiation equation of state with $e=4$ gives rise to
expressions like

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

The limits as $w^{*}\rightarrow 0$ are well defined and yield the
corresponding functions in the pure radiation case. If $\rho>0$ and bounded as
$\Omega\rightarrow 0$ we can assume for small $|\Omega|$ that

$w^{\prime}=\frac{1}{3}-\Omega^{4\,k}\,\rho^{k}\left((k+1)\,w^{*}+\Omega^{4}\,\rho\,(w^{*})^{\prime}\right)>0.$
(4.2)

The structural equation are regular with no condition on $k$. Equations
(3.15), (3.16), (3.17) are immediately seen to be regular with
$4\,k\in\mathbb{N}_{0}$. The fluid equation (3.10) reads

$\nabla_{0}\rho+\rho\left\\{\left(\frac{4}{3}-(\Omega^{4}\,\rho)^{k}\,w^{*}\right)\,\chi_{a}\,^{a}+3\,\Omega^{4\,k-1}\rho^{k}\,w^{*})\,\Sigma_{0}\right\\}=0.$
(4.3)

It is regular if $4\,k\in\mathbb{N}$. As long as the term in curly brackets is
continuous, the function $\rho$ will stay positive if it positive on some
Cauchy surface. With (4.2) the constraint (3.11) can be solved for
$\nabla_{a}\,\rho$ in a neighbourhood of $\Omega=0$

$\nabla_{a}\,\rho=-\rho\,\frac{4-3\,(\Omega^{4}\,\rho)^{k}\,w^{*}}{3\,w^{\prime}}\,f_{a}-\Omega^{4\,k-1}\,\rho^{k+1}\,\frac{(4\,k+3)\,w^{*}+4\,\Omega^{4}\,\rho\,(w^{*})^{\prime}}{w^{\prime}}\,\Sigma_{a}.$
(4.4)

The equation is regular if $4\,k\in\mathbb{N}$. Inspection of the equations

$e_{0}(f_{a})-w^{\prime}\,e_{a}(\chi_{c}\,^{c})+\chi_{ac}\,f^{c}+(1-3\,w^{\prime})\,L_{a0}-w^{\prime}\,\chi_{c}\,^{c}\,f_{a}=$
(4.5)
$=(1-3\,w^{\prime})\,(\Omega^{-1}\,\Sigma_{0}\,f_{a}+\Omega^{-1}\,\chi_{ab}\,\Sigma^{b}-\Omega^{-2}\,\Sigma_{0}\,\Sigma_{a})$
$-w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}\,(\chi_{c}\,^{c}\,f_{a}-3\,\Omega^{-1}\,\Sigma_{0}\,f_{a}-\Omega^{-1}\,\chi_{c}\,^{c}\,\Sigma_{a}+\Omega^{-2}\,\Sigma_{0}\,\Sigma_{a}),$

and

$e_{0}(\chi_{[ab]})-\chi_{c[a}\,\chi_{b]}\,^{c}-w^{\prime}\,\chi_{c}\,^{c}\,\chi_{[ab]}=(1-3\,w^{\prime})\,\Omega^{-1}\,\Sigma_{0}\,\chi_{[ab]},$
(4.6)

shows that they are regular if $4\,k-1\in\mathbb{N}$. With the expressions
above the equations

$6\,\Omega\,s-3\,\Sigma_{k}\,\Sigma^{k}-\lambda=\frac{1}{4}\,(\hat{\rho}-3\,w),$
(4.7)
$\nabla_{k}\,\Sigma_{j}=-\Omega\,L_{kj}+s\,g_{kj}+\Omega^{-1}\,\frac{\hat{\rho}+w}{2}\left(U_{k}\,U_{j}+\frac{1}{4}\,g_{kj}\right),$
(4.8) $\nabla_{k}\,s+\Sigma^{i}\,L_{ik}=$ (4.9)
$\Omega^{-2}\,\frac{\hat{\rho}+w}{2}\left(U_{k}\,U^{i}\,\Sigma_{i}+\frac{1}{4}\,\Sigma_{k}\right)+\frac{1}{24}\,(1-3\,w^{\prime})\,\Omega^{3}\left\\{\nabla_{k}\,\rho+4\,\rho\,\Omega^{-1}\,\Sigma_{k}\right\\},$

are seen to be regular if $4\,k\in\mathbb{N}_{0}$. We have finally

$\nabla_{i}\,L_{jk}-\nabla_{j}\,L_{ik}-\Sigma_{l}\,W^{l}\,_{kij}=\Omega\,M_{ijk},$
(4.10) $\nabla_{l}\,W^{l}\,_{kij}=M_{ijk},$ (4.11)

where

$M_{ijk}=2\,\Omega^{-1}\,\hat{\nabla}_{[i}\,\hat{L}_{j]k}=\Omega^{-4}\,(\hat{\rho}+w)\left(U_{k}\,U_{[i}\,\Sigma_{j]}-g_{k[i}\,U_{j]}\,\Sigma_{0}\right)$
(4.12)
$+4\,\rho\,\left((1+w^{\prime})\,\Sigma_{[i}\,U_{j]}\,U_{k}+\frac{1}{3}\,\Sigma_{[i}\,g_{j]k}\right)$
$+\Omega^{-3}\,(\hat{\rho}+w)\left(\nabla_{[i}\,U_{j]}\,U_{k}+U_{[j}\,\nabla_{i]}\,U_{k}\right)$
$+\Omega\,\left((1+w^{\prime})\,\nabla_{[i}\,\rho\,\,U_{j]}\,U_{k}+\frac{1}{3}\,\nabla_{[i}\,\rho\,g_{j]k}\right).$

Its components, given by

$M_{0b0}=\Omega^{-4}\,\frac{\hat{\rho}+w}{2}\,(\Sigma_{b}-\Omega\,f_{b})-\frac{2+3\,w^{\prime}}{6}\,(\Omega\,\nabla_{b}\,\rho+4\,\rho\,\Sigma_{b}),\quad$
(4.13)
$M_{ab0}=-\Omega^{-3}\,(\hat{\rho}+w)\,\chi_{[ab]}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,$
(4.14)
$M_{0bc}=\left\\{-\Omega^{-4}\,\frac{\hat{\rho}+w}{2}\,\Sigma_{0}+\frac{1}{6}\,(\Omega\,\nabla_{0}\rho+4\,\rho\,\Sigma_{0})\right\\}g_{bc}\quad\quad\quad\quad\,\,\,$
(4.15)
$M_{abc}=\frac{1}{3}\left\\{\Omega\,\nabla_{[a}\,\rho\,g_{b]c}+4\,\rho\,\Sigma_{[a}\,g_{b]c}\right\\},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,$
(4.16)

are regular if $4\,k\in\mathbb{N}_{0}$.

##  5 Cauchy problems.

To solve the conformal equations with an asymptotic radiation equation of
state we extract from the complete system a symmetric hyperbolic reduced
system for the unknown

${\bf Z}=(\Omega,\quad\Sigma_{k},\quad s,\quad
e^{\mu}\,_{a},\quad\Gamma_{a}\,^{b}\,{}_{c},\quad\rho,\quad\chi_{ab},\quad
f_{a},\quad L_{jk},\quad C^{i}\,_{jkl}),$

where the Ricci scalar $R=R[g]=6\,L_{j}\,^{j}$ of $g_{\mu\nu}$ plays the role
of a conformal gauge source function that controls implicitly the evolution of
the conformal factor. It can be prescribed as an arbitrary function of the
coordinates. The present discussion is concerned with the transition from the
‘physical part of the solution’ to scri and beyond. In physical terms this
involves a discussion of a domain of infinite temporal extent but in conformal
terms it involves only a finite step in the conformal time $\tau$. The
specification of $R[g]$ is thus not particularly critical. It is chosen to be
constant.

Symmetric hyperbolic reduced equations.

The reduced system is essentially built from the equations which contain
derivatives in the direction of $U$. However, sometimes these equations are
modified by using constraints, i.e. equations which do not contain derivatives
in the direction of $U$. For the first six components of Z we get the system

$e_{0}(\Omega)=\Sigma_{0},$
$e_{0}(\Sigma_{0})-f^{c}\,\Sigma_{c}=-\Omega\,L_{00}-s+\Omega^{-1}\,\frac{3\,(\hat{\rho}+w)}{8},$
$e_{0}(\Sigma_{a})-f_{a}\,\Sigma_{0}=-\Omega\,L_{0a},$
$\nabla_{0}\,s+\Sigma^{i}\,L_{i0}=\frac{1}{8}\,\Omega^{-2}\left\\{(1-3\,w^{\prime})\,w-3\,(\hat{\rho}+w)\right\\}\Sigma_{0}$
$-\frac{1}{24}\,(1-3\,w^{\prime})\,(\Omega^{3}\,\rho+\Omega^{-1}\,w)\,\chi_{a}\,^{a},$

where (3.10) resp. (4.3) has been used to replace $\nabla_{0}\rho$,

$e^{\mu}\,_{a,\,0}=-\chi_{a}\,^{b}\,e^{\mu}\,_{b}+f_{a}\,\delta^{\mu}\,_{0},$
$\Gamma_{a}\,^{b}\,{}_{c,\,0}=-f^{b}\,\chi_{ac}+\chi_{a}\,^{b}\,f_{c}-\chi_{a}\,^{d}\,\Gamma_{d}\,^{b}\,{}_{c}+\Omega\,W^{b}\,_{c0a}-g^{b}\,_{a}\,L_{0c}+L^{b}\,_{0}\,g_{ac},$

$\nabla_{0}\rho+(\rho+\Omega^{-4}\,w)\,\chi_{a}\,^{a}+\Omega^{-1}\,(\rho-3\,\Omega^{-4}\,w)\,\Sigma_{0}=0,$

Adding a suitable contraction of (3.7) to equation (4.5) gives the equation

$e_{0}(f_{a})-w^{\prime}\,e_{c}(\chi_{a}\,^{c})=-\chi_{ac}\,f^{c}-(1-3\,w^{\prime})\,L_{a0}+w^{\prime}\,\chi_{c}\,^{c}\,f_{a}$
(5.1)
$+(1-3\,w^{\prime})\,(\Omega^{-1}\,\nabla_{0}\Omega\,f_{a}+\Omega^{-1}\,\chi_{ab}\,\nabla^{b}\Omega-\Omega^{-2}\,\nabla_{0}\Omega\,\nabla_{a}\Omega)$
$-w^{\prime\prime}\,\frac{\hat{\rho}+w}{w^{\prime}}\,(\chi_{c}\,^{c}\,f_{a}-3\,\Omega^{-1}\,\nabla_{0}\,\Omega\,f_{a}-\Omega^{-1}\,\chi_{c}\,^{c}\,\nabla_{a}\Omega+\Omega^{-2}\,\nabla_{0}\Omega\,\nabla_{a}\Omega),$
$+w^{\prime}\,(2\,\chi_{[c}\,^{e}\,\Gamma_{a]e}\,^{c}-2\,f^{c}\,\chi_{ca}-2\,\Gamma_{[c}\,^{e}\,{}_{a]}\,\chi_{e}\,^{c}-2\,L^{0}\,_{a}).$

Adding (4.6) to (3.9) gives

$e_{0}(\chi_{ba})-e_{a}(f_{b})=-f_{d}\,\Gamma_{a}\,^{d}\,{}_{b}+f_{a}\,f_{b}-\chi_{a}\,^{d}\,\chi_{db}+\Omega\,W^{0}\,_{b0a}+L_{ab}+L^{0}\,_{0}\,g_{ab},$
(5.2)
$-2\,(\chi_{c[a}\,\chi_{b]}\,^{c}+w^{\prime}\,\chi_{c}\,^{c}\,\chi_{[ab]}+(1-3\,w^{\prime})\,\Omega^{-1}\,\Sigma_{0}\,\chi_{[ab]}).$

If equations (5.1) and (5.2) are written as a system for $f_{a}$ and
$\chi_{bc}$ with principal part

$\frac{1}{w^{\prime}}\,e_{0}(f_{a})-e_{c}(\chi_{a}\,^{c})=\ldots\,,\quad\quad
e_{0}(\chi_{ba})-e_{a}(f_{b})=\ldots\,.$

they represent, with given right hand sides, a symmetric hyperbolic system.

To derive the reduced systems for the curvature fields we follow the methods
of [4] and earlier articles. Equation (4.10) provides a system for the
unknowns $L_{0a}=L_{a0}$ and $L_{ab}=L_{ba}$ that contains no derivatives of
$L_{00}$. Where this quantity appears in the equation the contraction
$-L_{00}+g^{ab}\,L_{ab}=L_{j}\,^{j}=R/6$ can be used to express it in terms of
$L_{ab}$ and the gauge source function $R=R[g]=const$. The system is given by

$\nabla_{0}\,L_{0a}-g^{bc}\,\nabla_{b}\,L_{ac}=\Omega\,(M_{0a0}-g^{bc}\,M_{bac}),\quad
a=1,2,3,$ (5.3)
$\nabla_{0}\,L_{aa}-\nabla_{a}\,L_{0a}=\Sigma_{l}\,W^{l}\,_{a0a}+\Omega\,M_{0aa},\quad\,\,\,\quad
a=1,2,3,$ (5.4)
$2\,\nabla_{0}\,L_{ab}-\nabla_{a}\,L_{0b}-\nabla_{b}\,L_{0a}=-2\,\Sigma_{l}\,W^{l}\,_{(ab)0}+2\,\Omega\,M_{0(ab)},\,\,\,$
(5.5) $a,b=1,2,3,\,\,a<b.$

To extract the desired system from (4.11) consider the fields

$h^{j}\,_{k}=g^{j}\,_{k}+U^{j}\,U_{k},\,\,\,l^{j}\,_{k}=g^{j}\,_{k}+2\,U^{j}\,U_{k},\,\,\,\epsilon_{ijkl}=\epsilon_{[ijkl]},\,\,\,\epsilon_{jkl}=U^{i}\,\epsilon_{ijkl},\,\,\,\mbox{with}\,\,\,\epsilon_{0123}=1.$

The symmetric, trace-free $U$-electric part $w_{jl}$ and the $U$-magnetic part
$w^{*}_{jl}$ of $W^{i}\,_{jkl}$,

$w_{jl}=W_{ipkq}\,U^{i}\,h^{p}\,_{j}\,U^{k}\,h^{q}\,_{l},\quad\quad
w^{*}_{jl}=1/2\,W_{ipmn}\,\epsilon^{mn}\,_{kq}\,U^{i}\,h^{p}\,_{j}\,U^{k}\,h^{q}\,_{l},$

allow us to represent the rescaled conformal Weyl tensor in form

$W_{ijkl}=2\,(l_{i[k}\,w_{l]j}-l_{j[k}\,w_{l]i}-U_{[k}\,w^{*}_{l]p}\,\epsilon^{p}\,_{ij}-U_{[i}\,w^{*}_{j]p}\,\epsilon^{p}\,_{kl}).$

Suitable evolution equations for $w_{jl}$ and $w^{*}_{jl}$ is given by
$\nabla^{i}\,W_{i(a|dc|}\,\epsilon_{b)}\,^{dc}=K_{dc(a}\,\epsilon_{b)}\,^{dc}$,
$\nabla^{i}\,W_{i(a|0|b)}=K_{0(ab)}$ which read in detail

$e_{0}\,(w^{*}_{ab})-D_{d}\,w_{c(a}\,\epsilon_{b)}\,^{dc}=-\chi_{ed}\,w^{*}_{cf}\,\epsilon_{(a}\,^{fe}\,\epsilon_{b)}\,^{dc}\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(5.6)
$\quad\quad\quad\quad+(\chi^{e}\,_{(a}\,w^{*}_{b)e}-\chi\,w^{*}_{ab})+f_{d}\,w_{c(a}\,\epsilon_{b)}\,^{dc},$

and

$e_{0}(w_{ab})+D_{d}\,w^{*}_{c(a}\,\epsilon_{b)}\,^{dc}=-2\,\chi\,w_{ab}+2\,\chi^{c}\,_{(a}\,w_{b)c}\quad\quad\quad\quad\quad\quad\quad\quad$
(5.7)
$\quad\quad\quad\quad+\chi_{(a}\,^{c}\,w_{b)c}-\chi^{cd}\,w_{cd}\,\,g_{ab}+2\,w^{*}_{(a}\,{}^{e}\,\epsilon_{b)ed}\,f^{d}-M_{0(ab)},$

with ‘spatial derivatives’ $D_{a}\,w^{*}_{bc}\equiv
e_{a}\,(w^{*}_{bc})-\Gamma_{a}\,^{e}\,{}_{b}\,w^{*}_{ec}-\Gamma_{a}\,^{e}\,{}_{c}\,w^{*}_{be}=\nabla_{a}\,w^{*}_{bc}$.
The components of $M_{jkl}$ on the right hand sides of equations (5.3) - (5.7)
are given by

$M_{0a0}-g^{bc}\,M_{bac}=\Omega^{-4}\,\frac{\hat{\rho}+w}{2}\,(\Sigma_{a}-\Omega\,f_{a})-\frac{w^{\prime}}{2}\,(\Omega\,\nabla_{a}\,\rho+4\,\rho\,\Sigma_{a}),$
$M_{0ab}=-\left\\{\Omega^{-4}\,\frac{\hat{\rho}+w}{2}\,\Sigma_{0}-\frac{1}{6}\,(\Omega\,\nabla_{0}\rho+4\,\rho\,\Sigma_{0})\right\\}g_{ab},\quad\quad
M_{dc(a}\,\epsilon_{b)}\,^{dc}=0,$

where here and in (5.3), (5.4), (5.5) equations (4.3) and (4.4) must be used
to replace $\nabla_{0}\rho$ and $\nabla_{a}\rho$.

Observing the symmetry of $w_{ab}$, $w^{*}_{ab}$ (but ignoring their trace-
freeness, which can later be recovered as a consequence of the initial data
and the equations above), equations (5.6) and (5.7) can written as system for
the unknowns $w_{ab}$ and $w^{*}_{ab}$ with $a\leq b$, $a,b=1,2,3$. If the
equations $w_{ab,\,0}=\ldots$, $w^{*}_{ab,\,0}=\ldots$ are then written with a
factor 2 if $a<b$, the system is seen to be manifestly symmetric hyperbolic.

Together with the previous equations we have obtained now a quasi-linear
system for the unknown Z which can be written in the form ${\bf
A}^{\mu}\,\partial_{\mu}{\bf Z}={\bf B}$ with matrix-valued functions ${\bf
B}={\bf B}({\bf Z})$ and ${\bf A}^{\mu}={\bf A}^{\mu}({\bf Z})$ that are
symmetric, i.e. ${}^{t}{\bf A}^{\mu}={\bf A}^{\mu}$, with ${\bf A}^{0}$
defining a positive definite bilinear form if the $e^{0}\,_{a}$ are not too
large.

Cauchy data for the conformal equations.

Solutions that admit smooth conformal extensions at future time-like infinity
can be constructed from data for the conformal field equations on a compact
Cauchy hypersurface $S$ in the ‘physical domain’, where $\Omega>0$, or on the
compact 3-manifold $S={\cal J}^{+}=\\{\Omega=0\\}$ that represents future
time-like infinity.

Unless the field $U$ is assumed to be orthogonal to $S$, the data, which must
satisfy the constraints induced by the conformal field equations on $S$, are
in general expressed in terms of the unit normal to $S$ and then transformed
into a frame $e_{k}$ with $e_{0}=U$. This requires calculations involving the
complete system of conformal field equations which are fairly tedious (see [6]
where the presence of a boundary requires this) and give little insight. The
calculation will be skipped here.

We will first construct asymptotic end data on ${\cal J}^{+}$ with the
assumption that $U$ is orthogonal to ${\cal J}^{+}$. It is interesting to note
that in the case of a pure dust equation of state one is forced into this
requirement without forcing the field $U$ to be hypersurface orthogonal in the
physical domain (see [4]). In the present case this assumption implies a
genuine restriction. The analysis of the asymptotic end data follows
essentially the one given in the vacuum case [1] with some modifications if
matter fields are involved [friedrich:massive-fields], [4]

To analyse the constraints, i.e. the equations that do not involve derivatives
in the direction of $U$, we assume that $k\geq 1$. The restrictions of the
equations to ${\cal J}^{+}$ are then regular and the expressions containing
the quantities $w^{*}$, $(w^{*})^{\prime}$ drop out. The unknowns are then
determined as follows.

The requirement that $U$ is orthogonal to ${\cal J}^{+}$ implies

$e^{0}\,_{a}=0,\quad\Sigma_{a}=0,\quad\chi_{ab}\,\,\,\mbox{is the second
fundamental form induced on ${\cal J}^{+}$}.$

The first structural equation (3.1) implies on ${\cal J}^{+}$ the first
structural equation with respect to the the frame $e_{a}$ and metric $h_{ab}$
induced on ${\cal J}^{+}$. The $\Gamma_{a}\,^{c}\,{}_{b}$ are thus the
connection coefficients of the covariant derivative operator $D$ induced by
the metric $h_{ab}$ in the frame $e_{a}$. The second structural equation (3.2)
reduces to the relation
$R^{a}\,_{cdb}[h]=2\,g^{a}\,_{[d}\,L_{b]c}+2\,L^{a}\,_{[d}\,g_{b]c}$,
equivalent to

$L_{ab}[g]=l_{ab}[h]\quad\mbox{with}\quad
l_{ab}[h]=R_{ab}[h]-\frac{1}{4}\,R[h]\,h_{ab}.$

This gives $L_{00}=g^{ab}\,L_{ab}-L_{j}\,^{j}=R[h]/4-R/6$ where $R=R[g]$
represents the conformal gauge source function. Equations (3.11), (3.15),
(3.16), (3.17) imply

$f_{a}=-\frac{1}{4}\,\nabla_{a}\rho,\quad\Sigma^{0}=\nu\equiv\sqrt{\lambda/3}>0,\quad\chi_{ab}=s/\nu\,g_{ab},\quad\nabla_{a}\,s=-\nu\,L_{0a},$

where it is assumed that $U$ is future directed and $\Omega$ is decreasing in
the direction of $U$. By a rescaling
$g_{\mu\nu}\rightarrow\theta^{2}\,g_{\mu\nu}$ and
$\Omega\rightarrow\theta\,\Omega$, where $\theta>0$ is smooth with prescribed
value and suitable normal derivative on ${\cal J}^{+}$, it can be achieved
that

$s=0\quad\mbox{whence}\quad\chi_{ab}=0,\quad
L_{0a}=0\quad\quad\mbox{on}\quad{\cal J}^{+}.$

Equation (3.18), which reduces to

$D_{c}\,l_{da}\,\epsilon_{b}\,^{cd}=\nu\,w^{*}_{ab},$

relates the Cotton tensor of $h$, given on the left hand side, to the
$U$-magnetic part of $W^{i}\,_{jkl}$. The constraints induced by (3.19), given
by $\nabla_{l}\,W^{l}\,_{0ij}=M_{ij0}$, translate with $M_{ab0}=0$ and
$M_{0b0}=0$ to the constraint

$D^{a}\,w_{ab}=0,$ (5.8)

and to $D^{a}\,w^{*}_{ab}=0$, which is not a constraint but the identity
satisfied by the Cotton tensor.

If the data

$e^{\alpha}\,_{a},\quad\rho,\quad w_{ab},$

are prescribed on ${\cal J}^{+}$ so that
$h^{\alpha\beta}=\delta^{ab}\,e^{\alpha}\,_{a}\,e^{\beta}\,_{b}$ defines the
contravariant version of a Riemannian metric, $\rho>0$, and $w_{ab}=w_{ba}$,
$w_{c}\,^{c}=0$ with $w_{ab}$ satisfying (5.8) with the covariant derivative
operator $D$ defined by $h$, all the unknowns subsumed by ${\bf Z}$ can be
determined in our gauge.

On the preservation of the constraints.

Smooth data on ${\cal J}^{+}$ as described above, determine a smooth solution
to the symmetric hyperbolic reduced equations that cover a neighbourhood of
${\cal J}^{+}=\\{\tau=0\\}$ with $\Omega>0$ in the past of ${\cal J}^{+}$,
where the parameter $\tau$ on the flow lines of $U$ is negative, and
$\Omega<0$ in the future of ${\cal J}^{+}$ where $\tau>0$ [9]. We can assume
it to exist in a range $-\tau_{*}\leq\tau\leq\tau_{*}$ with some $\tau_{*}>0$
and $\tau=0$ on ${\cal J}^{+}$.

Once this solution has been obtained, it remains to show that the solution to
the reduced equations does in fact also satisfy the constraints and thus the
complete system of conformal equations. Since the proof follows a standard
recipe which has been worked out in detail in earlier articles (see [4], and
in particular [3], [7] for a discussion of the Einstein-perfect-fluid
equations with a general equation of state), we skip this step.

There remains, however, an open question. Since we assumed $U$ to be
orthogonal to ${\cal J}^{+}$ we should expect $U$ to be hypersurface
orthogonal, that is $\chi_{[ab]}=0$ in the range
$-\tau_{*}\leq\tau\leq\tau_{*}$. Because $\chi_{[ab]}=0$ on ${\cal J}^{+}$,
one could think this to be a consequence of equation (4.6). Since this
equation is neither a constraint nor satisfied as part of the reduced system,
we use a different argument. The evolution of $\chi_{ab}$ is governed by
equation (5.2), which implies

$e_{0}(\chi_{[ba]})=D_{[a}f_{b]}+\chi_{[ac]}\,\beta_{b}\,^{c}-\chi_{[bc]}\,\beta_{a}\,^{c}-2\,w^{\prime}\,\chi_{c}\,^{c}\,\chi_{[ab]}-2\,(1-3\,w^{\prime})\,\Omega^{-1}\,\Sigma_{0}\,\chi_{[ab]},$

with $\beta_{ab}=\chi_{(ac)}$.We formally write here
$D_{a}f_{b}=e_{a}(f_{b})-f_{d}\,\Gamma_{a}\,^{d}\,{}_{b}$, not implying that
$D$ denotes a covariant derivative away from ${\cal J}^{+}$. Among the
constraints which we can assume to be satisfied there is the relation (3.11)
or its more concise version (3.12), i.e.

$f_{a}=e_{a}(f)\quad\mbox{with}\quad f=F+\log\Omega\quad\mbox{and}\quad
F^{\prime}=-(\hat{\rho}+w)^{-1}\,w^{\prime}.$ (5.9)

Since the connection $\nabla$ is torsion free it follows

$0=\nabla_{a}\nabla_{b}f-\nabla_{b}\nabla_{a}f=2\,(D_{[a}f_{b]}-\chi_{[ab]}\,e_{0}(f)),$

where the function

$e_{0}(f)=e_{0}(F+\log\Omega)=-\frac{w^{\prime}}{\rho+\Omega^{-4}\,w}\,\nabla_{0}\rho+\Omega^{-1}\,\frac{\rho+\Omega^{-4}\,w-4\,\rho\,w^{\prime}}{\rho+\Omega^{-4}\,w}\,\nabla_{0}\Omega,$

is regular if $k\geq 1$. The resulting ODE for $\chi_{[ab]}$ implies that
$\chi_{[ab]}=0$, equation (4.6) is satisfied, and $U$ is hypersurface
orthogonal in the range $-\tau_{*}\leq\tau\leq\tau_{*}$.

With this we discussed all the ingredients needed to obtain existence results.
General results in symmetric hyperbolic systems [9] allow us to draw the
following conclusions.

Let the reduced conformal Einstein-$\lambda$-perfect-fluid equations with
asymptotic radiation equation of state where $k\geq 1$ be given in terms of
the gauge discussed above so that $e_{0}=U$. Assume the future directed flow
vector $U$ to be orthogonal to the compact 3-manifold ${\cal
J}^{+}=\\{\Omega=0\\}$. Let on ${\cal J}^{+}$ be given a minimal end data set
consisting of a smooth frame field
$e_{a}=e^{\alpha}\,_{a}\,\partial_{x^{\alpha}}$, a positive function $\rho$,
and a symmetric trace-free tensor field $w_{ab}$ that satisfies
$D^{a}\,w_{ab}=0$ where $D$ is the covariant derivative operator associated
with the metric $h$ that satisfies $h(e_{a},e_{b})=\delta_{ab}$. Then:

$-$ As discussed above, a complete set of Cauchy data for the conformal field
equations can be calculated from the minimal set if the gauge conditions, the
constraints, and the special features of ${\cal J}^{+}=\\{\Omega=0\\}$ are
taken into account.

$-$ These data determine a smooth solution $\Omega,\,g,\,\ldots,\,$ to the
conformal field equation (unique up to extensions) so that $\Omega<0$ in the
future of ${\cal J}^{+}$, $\Omega>0$ in the past of ${\cal J}^{+}$, and
$\hat{g}=\Omega^{-2}\,g$, $\hat{U}=\Omega\,U$, $\hat{\rho}=\Omega^{4}\,\rho$
satisfy where $\Omega\neq 0$ the Einstein-$\lambda$-perfect-fluid equations
with asymptotic radiation equation of state. The flow field $\hat{U}$ is
hypersurface orthogonal.

$-$ Let $S$ by a Cauchy hypersurface in the ‘physical domain’ of this solution
where $\Omega>0$ and denote by $\Delta$ the Cauchy data induced by the
solution on $S$. Let $\Delta^{\prime}$ be Cauchy data on $S$ for the same
system of equations. If these data are sufficiently close to $\Delta$ they
develop into a solution which is time-like and null geodesically future
complete, admits a smooth conformal boundary ${\cal J}^{\prime+}$ representing
its future time-like infinity and a smooth conformal extension beyond with
$\rho>0$.

$-$ If the cases where $w^{*}=0$ is admitted, the resulting set of solutions
contains as special cases the FLRM-pure-radition solutions which develop a
smooth conformal boundary in the future.

Given the data $\Delta^{\prime}$ whence the corresponding flow field
$U^{\prime}$ on $S$, the field equations allow us to determine the quantity
$U^{\prime}_{[i}\,\nabla^{\prime}_{j}U^{\prime}_{k]}$ on $S$. In the last
statement it is not required that this quantity vanishes on $S$. Thus the flow
field $U^{\prime}$ determined by the data $\Delta^{\prime}$ will not
necessarily be hypersurface orthogonal and $U^{\prime}$ need not be orthogonal
to ${\cal J}^{\prime+}$.

If Cauchy data are given on ${\cal J}^{+}$ with a flow vector field $U$ that
is not necessarily orthogonal to ${\cal J}^{+}$ the result stated at the end
of the introduction is obtained.

## References

  * [1] H. Friedrich.  Existence and structure of past asymptotically simple solution of Einstein’s field equations with positive cosmological constant.  J. Geom. Phys. 3 (1986) 101 - 117.
  * [2] H. Friedrich.  On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure.  Commun. Math. Phys. 107 (1986), 587 - 609.
  * [3] Friedrich, H. (1998)  Evolution equations for gravitating ideal fluid bodies in general relativity.  Phys. Rev. D 57, 2317–2322.
  * [4] H. Friedrich.  Sharp asymptotics for Einstein-$\lambda$-dust flows.  Commun. Math. Phys. (2016) DOI 10.1007/s00220-016-2716-6.  arXiv:1601.04506
  * [5] H. Friedrich.  Hierarchies of asymptotic behaviour in cosmological models with positive $\lambda$.  In preparation.
  * [6] H. Friedrich, G. Nagy.  The initial boundary value problem for Einstein’s vacuum field equations.  Comm. Math. Phys. 201 (1999) 619 - 655.
  * [7] H. Friedrich, A. Rendall.  The Cauchy Problem for the Einstein Equations.  In: B. Schmidt (ed.): Einstein’s Field Equations and Their Physical Implications. Springer, Lecture Notes in Physics, Berlin 2000.
  * [8] M. Had$\check{z}$ić, J. Speck.  The global future stability of the FLRW solutions to the Dust–Einstein system with a positive cosmological constant.  J. Hyperbolic Differential Equations 12 (2015) 87.
  * [9] T. Kato.  The Cauchy problem for quasi-linear symmetric hyperbolic systems.  Arch. Ration. Mech. Anal., 58 (1975) 181 - 205.
  * [10] C. Liu, C. Wei.  Future stability of the FLRW space-time for a large class of perfect fluids.  Ann. Henri Poincaré, 22 (2021) 715 - 770.
  * [11] C. Lübbe, J. A. Valiente Kroon.  A conformal approach to the analysis of the non-linear stability of radiation cosmologies.  Ann. Phys. 328 (2013) 1 - 25.
  * [12] T. A. Olyinyk.  Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant.  Commun. Math. Phys 346 (2016) 293 - 312
  * [13] O. Reula.  Exponential decay for small non-linear perturbations of expanding flat homogeneous cosmologies.  Phys. Rev. D 60 (1999) 083507.
  * [14] I. Rodnianski, J. Speck.  The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant. J. Eur. Math. Soc. 15 (2013) 2369 - 2462.
  * [15] J. Speck.  The nonlinear future stability of the FLRW family of solutions to the Euler-Einstein system with a positive cosmological constant. Sel. Math. New Ser. 18 (2012) 633 - 715.