# Radiative Poincare type eon and its follower

Paweł Nurowski  Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al.
Lotników 32/46, 02-668 Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract.

We consider two consecutive eons $\hat{M}$ and $\check{M}$ from Penrose’s
Conformal Cyclic Cosmology and study how the matter content of the past eon
($\hat{M}$) determines the matter content of the present eon ($\check{M}$) by
means of the reciprocity hypothesis.

We assume that the only matter content in the final stages of the past eon is
a spherical wave described by Einstein’s equations with the pure radiation
energy momentum tensor

$\hat{T}^{ij}=\hat{\Phi}K^{i}K^{j},\quad\hat{g}_{ij}K^{i}K^{j}=0,$

and with cosmological constant $\hat{\Lambda}$ . We solve these Einstein’s
equations associating to $\hat{M}$ the metric $\hat{g}=t^{-2}\big{(}-{\rm
d}t^{2}+h(t)\big{)}$, which is a Lorentzian analog of the Poincaré-Einstein
metric known from the theory of conformal invariants. The solution is obtained
under the assumption that the 3-dimensional conformal structure $[h]$ on the
$\mathscr{I}^{+}$ of $\hat{M}$ is flat, that the metric $\hat{g}$ admits a
power series expansion in the time variable $t$, and that $h(0)\in[h]$. Such
solution depends on one real arbitrary function of the radial variable $r$.

Applying the reciprocal hypothesis, $\hat{g}\to\check{g}=t^{4}\hat{g}$, we
show that the new eon $(\check{M},\check{g})$ created from the one containing
a single spherical wave, is filled at its initial state with three types of
radiation: (i) the damped spherical wave which continues its life from the
previous eon, (ii) the in-going spherical wave obtained as a result of a
collision of the wave from the past eon with the Bang hypersurface and (3)
randomly scattered waves that could be interpreted as perfect fluid with the
energy density $\check{\rho}$ and the isotropic pressure $\check{p}$ such that
$\check{p}=\tfrac{1}{3}\check{\rho}$. The metric $\check{g}$ solves the
Einstein’s equations without cosmological constant and with the energy-
momentum tensor

$\check{T}^{ij}=\check{\Phi}K^{i}K^{j}+\check{\Psi}L^{i}L^{j}+(\check{\rho}+\check{p})\check{u}^{i}\check{u}^{j}+\check{p}\check{g}^{ij},$

in which $\check{u}^{i}\check{u}^{j}\check{g}_{ij}=-1$,
$\check{g}_{ij}L^{i}L^{j}=0$ and $L^{i}K^{j}\check{g}_{ij}=-2$.

The research was funded from the Norwegian Financial Mechanism 2014-2021 with
project registration number 2019/34/H/ST1/00636.

##  1\. The setting

In this short note we show a model of a bandage region of two consecutive eons
from the _Penrose’s Conformal Cyclic Cosmology_ (CCC) [2], which have the
following properties111We closely follow Paul Tod’s setting, notation and
terminology as presented in [3]:

  * •

The common three-surface $\Sigma$ of $\mathscr{I}^{+}$ of the past eon and the
Big Bang of the present eon is equipped with a conformal class $[h_{0}]$ of
signature $(+,+,+)$ which _has vanishing Cotton tensor_ , i.e. $[h_{0}]$ is
_conformally flat_ ; in the following we will chose $h_{0}$ to be the _flat_
representative of the conformal class $[h_{0}]$;

  * •

The Poincaré-type extension $(\hat{M},\hat{g})$, with
$\hat{M}=]0,\epsilon[\times\Sigma$, of the conformal three-manifold
$(\Sigma,[h_{0}])$ has the Lorentzian metric [1]:

(1.1) $\hat{g}=t^{-2}(-{\rm d}t^{2}+h_{t});$

Here $t\in]0,\epsilon[$ is a coordinate along the extension $\mathbb{R}_{+}$
of $\Sigma$ in $\hat{M}$, and $h_{t}=h(t,x)$ is a $t$-dependent 1-parameter
family of metrics on $\Sigma$ such that $h(0,x)=h_{0}\in[h_{0}]$; here $x$ is
a point in $\Sigma$, $x\in\Sigma$;

  * •

The Poincaré-type metric $\hat{g}$ in $\hat{M}$ satisfies the _pure radiation
Einstein’s equations with cosmological constant_ $\hat{\Lambda}$:

(1.2) $\hat{R}^{ij}=\hat{\Lambda}\hat{g}^{ij}+\hat{\Phi}K^{i}K^{j};$

Here $K^{i}$ is an _expanding null_ vector field _without shear and without
twist_ on $\hat{M}$; in particular we have $\hat{g}_{ij}K^{i}K^{j}=0$;

  * •

The Lorentzian four-metric

$g=-{\rm d}t^{2}+h_{t}$

conformal to the Poincare-type metric $\hat{g}$ in $\hat{M}$ is naturally
extended to $M=\hat{M}\cup\Sigma\cup\check{M}$, which is a bundle $\Sigma\to
M\stackrel{{\scriptstyle\pi}}{{\to}}I$ over the interval
$I=]-\epsilon,\epsilon[\subset\mathbb{R}$ parameterized by $t$, with the
following preimages of $]-\epsilon,\epsilon[$: $\pi^{-1}(t>0)=\hat{M}$,
$\pi^{-1}(t=0)=\Sigma$, and $\pi^{-1}(t<0)=\check{M}$;

  * •

The metric $g$ is used to define a Lorentzian metric $\check{g}$ in
$\check{M}$, which is

$\check{g}=t^{2}(-{\rm d}t^{2}+h_{t}),\quad\quad\mathrm{for}\quad t<0;$

  * •

Note that for $t>0$ we have $\hat{g}=\hat{\Omega}^{2}g$ and that for $t<0$ we
have $\check{g}=\check{\Omega}^{2}g$ with
$\check{\Omega}=-\hat{\Omega}^{-1}=t$.

One of the aims of this note is to identify the above four-manifold $M$,
equipped with the three Lorentzian metrics $\hat{g}$, $g$ and $\check{g}$,
with the _bandage region_ [3] of the _Penrose’s cyclic Universe_ [2] in which
the _past eon ends_ as filled _with only one spherical gravitational wave_
propagating along the null vector $K^{i}$. Forcing the Poincaré-type expansion
metric $\hat{g}$ to satisfy the Einstein’s equations (1.2) was the first step
to achieve this aim. Another aim is to see how the gravitational wave
contained in the past eon will change into a matter content at the beginning
of the present eon, by means of _Penrose’s reciprocal hypothesis_ , stating
that the three metrics $\hat{g}$, $g$ and $\check{g}$ in the bandage region
should be related via: $\hat{g}=\Omega^{-2}g$ and $\check{g}=\Omega^{2}g$.

As we show below, under further simplification assumptions, the explicit form
of the Poincaré-type metric $\hat{g}$ can be easily found up to an arbitrarily
prescribed accuracy, and as a byproduct one gets a _remarkably pleasant_
consequences of the so obtained $\hat{g}$ for $\check{g}$, and in particular
for the _matter content of the spacetime_ $(\check{M},\check{g})$, which is
interpreted as the beginning of the _present eon_.

##  2\. The ansatz and the model for the past eon

In the _theory of conformal invariants_ as presented by Fefferman and Graham
in [1], given a conformal class $[h]$ on $\Sigma$, one obtains the system of
conformal invariants of $[h]$ in terms of the (pseudo)Riemannian invariants of
a certain (pseudo)Riemannian metric $\hat{g}$. This metric is naturally
associated with the conformal class $[h]$ via $\hat{g}=t^{-2}(\varepsilon{\rm
d}t^{2}+h_{t})$, where $h_{t}$ is a 1-parameter family of metrics on $\Sigma$,
such that $h_{0}=h$ and $h$ is a representative of $[h]$. The metric $\hat{g}$
is defined for $t>0$ and the value $\varepsilon=1$ is chosen. To encode the
conformal properties of $[h]$, this metric is demanded to be _unique_. This is
done by the requirements that $\hat{g}$ is _Einstein_ ,
$\hat{Ric}(\hat{g})=\hat{\Lambda}\hat{g}$, and that $h_{t}$ is real analytic
and symmetric in $t$.

We present here a _milder version of this construction_ , applied to the
conformally flat Riemannian structure $(\Sigma,[h])$ on a three-dimensional
$\Sigma$, to obtain the metric $\hat{h}$ of the past eon with a desirable
physical properties. In our ‘milder version’ we do as follows:

  * (a)

We replace $\varepsilon=1$ by $\varepsilon=-1$ \- to have the Lorentzian
signature of $\hat{g}$;

  * (b)

We replace the _Einstein condition_ $\hat{Ric}(\hat{g})=\hat{\Lambda}\hat{g}$
by the Einstein equation (1.2) - to have the past eon filled by a spherical
gravitational wave;

  * (c)

And we drop the condition that $h_{t}$ is symmetric in the variable $t$ \- to
have more flexibility on the matter content of the present eon $\check{M}$.

Since in our milder-than-Fefferman-Graham-setting we do not have uniqueness
theorems as in [1] the outcome past eon metric $\hat{g}$ is not rigidly
constrained. Thus, instead of working with the most general form of the the
1-parameter family of metrics $h_{t}$ _we make a physically motivated ansatz_
for them, hoping that it is compatible with the Einstein’s equations (1.2).

We start with a conformal class $[h_{0}]$ represented by the flat
3-dimensional metric

$h_{0}=\frac{2r^{2}{\rm d}z{\rm d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}+{\rm
d}r^{2}.$

Then as $h_{t}$ we take the _spherically symmetric_ 1-parameter family

$h_{t}=\frac{2r^{2}\big{(}1+\nu(t,r)\big{)}{\rm d}z{\rm
d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}+\big{(}1+\mu(t,r)\big{)}{\rm d}r^{2},$

where the unknown function $\nu=\nu(t,r)$ and $\mu=\mu(t,r)$ are both _real
analytic_ in the variable $t$ and such that:

$\nu(0,r)=0\quad\mathrm{and}\quad\mu(0,r)=0.$

This obviously satisfies $h_{t=0}=h_{0}$ and because of the analyticity
assumption we have

$\nu(t,r)=\sum_{i=1}^{\infty}a_{i}(r)t^{i}\quad\mathrm{and}\quad\mu(t,r)=\sum_{i=1}^{\infty}b_{i}(r)t^{i},$

with a set of differentiable functions $a_{i}=a_{i}(r)$ and $b_{i}=b_{i}(r)$
depending on the $r$ variable only.

This leads to the following ansatz for the pre-Poincaré-type metric $\hat{g}$
in $\hat{M}$:

(2.1) $\hat{g}=t^{-2}\,\Big{(}\,-{\rm
d}t^{2}\,+\,\frac{2r^{2}\big{(}\,1+\sum_{i=1}^{\infty}a_{i}(r)t^{i}\,\big{)}{\rm
d}z{\rm
d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}\,+\,\big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\big{)}{\rm
d}r^{2}\,\Big{)}.$

Our (pre)past eon manifold $\hat{M}$ is parameterized by $t>0$, $r>0$ and
$z\in\mathbb{C}\cup\\{\infty\\}$.

We now consider the following null vector field $K$ on $\hat{M}$:

$K=\partial_{t}+\Big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\Big{)}^{-\tfrac{1}{2}}\partial_{r}.$

It is tangent to a congruence of null geodesics without shear and twist, which
represents light rays emanating from the source at the surface $r=0$. We
require that the Poincaré-type metric (2.1) satisfies the Einstein equations
(1.2) with this null vector field $K$ and some functions $\hat{\Phi}$ and
$\hat{\Lambda}$. We have the following theorem/conjecture.

Theorem 1.
If the metric

(2.2) $\displaystyle\hat{g}=$ $\displaystyle t^{-2}(-{\rm d}t^{2}+h_{t})=$
$\displaystyle t^{-2}\,\Big{(}\,-{\rm
d}t^{2}\,+\,\frac{2r^{2}\big{(}\,1+\nu(t,r)\,\big{)}{\rm d}z{\rm
d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}\,+\,\big{(}\,1+\mu(t,r)\,\big{)}{\rm
d}r^{2}\,\Big{)}=$ $\displaystyle t^{-2}\,\Big{(}\,-{\rm
d}t^{2}\,+\,\frac{2r^{2}\big{(}\,1+\sum_{i=1}^{\infty}a_{i}(r)t^{i}\,\big{)}{\rm
d}z{\rm
d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}\,+\,\big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\big{)}{\rm
d}r^{2}\,\Big{)}$

satisfies Einstein’s equations

(2.3)
$\hat{E}{}_{ij}:=\hat{R}{}_{ij}-\hat{\Lambda}\hat{g}{}_{ij}-\hat{\Phi}\hat{K}{}_{i}\hat{K}{}_{j}=0$

with

(2.4)
$K=K^{i}\partial_{i}=\partial_{t}+\Big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\Big{)}^{-\tfrac{1}{2}}\partial_{r},\quad\quad\hat{K}_{i}=\hat{g}_{ij}K^{j},$

then we have:

  * •

The coefficients $a_{1}(r)$, $a_{2}(r)$ $b_{1}(r)$ and $b_{2}(r)$ identically
vanish, $a_{1}(r)=a_{2}(r)=b_{1}(r)=b_{2}(r)=0$, and the power series
expansion of $h_{t}$ starts at the $t^{3}$ terms,
$h_{t}=t^{3}\chi(r)+\mathcal{O}(t^{4})$.

  * •

The metric $\hat{g}$, or what is the same, the power series expansions
$\nu(t,r)=\sum_{i=1}^{\infty}a_{i}(r)t^{i}$ and
$\mu(t,r)=\sum_{i=1}^{\infty}b_{i}(r)t^{i}$, are totally determined up to
infinite order by an arbitrary differentiable function $f=f(r)$.

  * •

More precisely, the Einstein equations $\hat{E}{}_{ij}=\mathcal{O}(t^{k+1})$
solved up to an order $k$, together with an arbitrary differentiable function
$f=f(r)$, uniquely determine $\nu(t,r)$ and $\mu(t,r)$ up to an order $(k+2)$.

  * •

In the lowest order the solution reads:

$\nu=\frac{f}{r^{3}}t^{3}+\mathcal{O}(t^{4})\quad\mathrm{and}\quad\mu=-\frac{2f}{r^{3}}t^{3}+\mathcal{O}(t^{4});$

The energy function $\hat{\Phi}$ and the cosmological constant $\hat{\Lambda}$
are:

$\hat{\Phi}=3\frac{f^{\prime}}{r^{3}}t^{6}+\mathcal{O}(t^{7})\quad\mathrm{and}\quad\hat{\Lambda}=3+\mathcal{O}(t^{k+3});$

the Weyl tensor of the solution is

$\hat{W}^{i}{}_{jkl}=\mathcal{O}(t).$

In particular, the Weyl tensor $\hat{W}^{i}{}_{jkl}$ vanishes at $t=0$ and
$\hat{\Lambda}>0$ there.

With the use of computers we calculated this solution up to the order $k=10$,
finding explicitly $\nu=\sum_{k=3}^{10}a_{k}t^{k}$ and
$\mu=\sum_{k=3}^{10}b_{k}t^{k}$. The formulas are compact enough up to $k=8$
and up to the order $k=8$ they read:

$\displaystyle\nu(t,r)=$ $\displaystyle
f\tfrac{t^{3}}{r^{3}}\,-\,\tfrac{3}{4}f^{\prime}\tfrac{t^{4}}{r^{4}}+\tfrac{1}{10}\big{(}-2rf^{\prime}+3r^{2}f^{\prime\prime}\big{)}\tfrac{t^{5}}{r^{5}}+$
$\displaystyle\tfrac{1}{24}\big{(}3f^{2}-3rf^{\prime}+3r^{2}f^{\prime\prime}-2r^{3}f^{(3)}\big{)}\tfrac{t^{6}}{r^{6}}+$
$\displaystyle\tfrac{r}{280}\big{(}-24f^{\prime}-105ff^{\prime}+24rf^{\prime\prime}-12r^{2}f^{(3)}+5r^{3}f^{(4)}\big{)}\tfrac{t^{7}}{r^{7}}-$
$\displaystyle\tfrac{r}{960}\big{(}60f^{\prime}+288ff^{\prime}-150rf^{\prime}{}^{2}-60rf^{\prime\prime}-216rff^{\prime\prime}+30r^{2}f^{(3)}-10r^{3}f^{(4)}+3r^{4}f^{(5)}\big{)}\tfrac{t^{8}}{r^{8}}+$
$\displaystyle\mathcal{O}(\big{(}\tfrac{t}{r}\big{)}^{9})$
$\displaystyle\mu(t,r)=$
$\displaystyle-2f\tfrac{t^{3}}{r^{3}}\,+\,\tfrac{3}{4}f^{\prime}\tfrac{t^{4}}{r^{4}}-\tfrac{1}{5}f^{\prime\prime}\tfrac{t^{5}}{r^{5}}\,+\,\tfrac{1}{24}\big{(}39f^{2}+r^{3}f^{(3)}\big{)}\tfrac{t^{6}}{r^{6}}\,-\,\tfrac{r}{280}\big{(}390ff^{\prime}+2r^{3}f^{(4)}\big{)}\tfrac{t^{7}}{r^{7}}+$
$\displaystyle\tfrac{r}{960}\big{(}-18ff^{\prime}+300rf^{\prime}{}^{2}+378rff^{\prime\prime}+r^{4}f^{(5)}\big{)}\tfrac{t^{8}}{r^{8}}+\mathcal{O}(\big{(}\tfrac{t}{r}\big{)}^{9}).$

For a solution up to this order we find that:

$\displaystyle\hat{\Phi}\,=$ $\displaystyle
3r^{3}f^{\prime}\tfrac{t^{6}}{r^{6}}\,+\,3r^{3}\big{(}f^{\prime}-rf^{\prime\prime}\big{)}\tfrac{t^{7}}{r^{7}}\,+\,\tfrac{3r^{3}}{2}\big{(}2f^{\prime}-2rf^{\prime\prime}+r^{2}f^{(3)}\big{)}\tfrac{t^{8}}{r^{8}}\,+$
$\displaystyle\tfrac{r^{3}}{2}\big{(}6f^{\prime}+6ff^{\prime}-6rf^{\prime\prime}+3r^{2}f^{(3)}-r^{3}f^{(4)}\big{)}\tfrac{t^{9}}{r^{9}}+$
$\displaystyle\tfrac{r^{3}}{8}\big{(}24f^{\prime}+66ff^{\prime}-12rf^{\prime}{}^{2}-24rf^{\prime\prime}-30rff^{\prime\prime}+12r^{2}f^{(3)}-4r^{3}f^{(4)}+r^{4}f^{(5)}\big{)}\tfrac{t^{10}}{r^{10}}+$
$\displaystyle\tfrac{r^{3}}{40}\big{(}120f^{\prime}+522ff^{\prime}-177rf^{\prime}{}^{2}-120rf^{\prime\prime}-378rff^{\prime\prime}+93r^{2}f^{\prime}f^{\prime\prime}+60r^{2}f^{(3)}+90r^{2}ff^{(3)}-20r^{3}f^{(4)}+5r^{4}f^{(5)}-r^{5}f^{(6)}\big{)}\tfrac{t^{11}}{r^{11}}+$
$\displaystyle\mathcal{O}(\Big{(}\tfrac{t}{r}\Big{)}^{12}),$
$\hat{\Lambda}=3+\mathcal{O}(t^{9}).$

I have no patience to type the Weyl tensor components up to high order. It is
enough to say that that up to the 4th order in $t$, modulo a nonzero constant
tensor $C^{i}{}_{jkl}$, it is equal to:

$\hat{W}^{i}{}_{jkl}=\Big{(}\frac{f}{r^{2}}\frac{t}{r}-\frac{f^{\prime}}{r}\frac{t^{2}}{r^{2}}+\frac{f^{\prime\prime}}{2}\frac{t^{3}}{r^{3}}\Big{)}C^{i}{}_{jkl}+\mathcal{O}(\Big{(}\tfrac{t}{r}\Big{)}^{4}).$

Of course, for the positivity of the energy density $\hat{\Phi}$ close to the
surface $\mathscr{I}^{+}$ of $\hat{M}$ we need

$f^{\prime}>0.$

######  Corollary 2.1.

The Poincaré-type metric (2.2) can be interpreted as the ending stage of the
evolution of the past eon in Penrose’s CCC. The eon has a positive
cosmological constant $\hat{\Lambda}\simeq 3$, which is filled with a
spherically symmetric pure radiation moving along the null congruence
generated by the vector field $K$.

##  3\. Using reciprocity for the model of the present eon

Now, following the Penrose-Tod reciprocal hypothesis procedure, we summarize
the properties of the spacetime $\check{M}$ equipped with the metric
$\check{g}$ obtained from $\hat{g}$ as in Theorem 1, by the reciprocal change
$\check{\Omega}\to-\hat{\Omega}^{-1}=t$. In other words, we are now interested
in the properties of the metric $\check{g}=t^{4}\hat{g}$. We have the
following theorem.

Theorem 2.
Assume that the metric $\hat{g}$ as in (2.2) satisfies the Einstein equations
(2.3)-(2.4), $\hat{E}{}_{ij}=0$. Then, the reciprocal metric

$\displaystyle\check{g}=$ $\displaystyle t^{2}\,\Big{(}\,-{\rm
d}t^{2}\,+\,\frac{2r^{2}\big{(}\,1+\nu(t,r)\,\big{)}{\rm d}z{\rm
d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}\,+\,\big{(}\,1+\mu(t,r)\,\big{)}{\rm
d}r^{2}\,\Big{)}=$ $\displaystyle t^{2}\,\Big{(}\,-{\rm
d}t^{2}\,+\,\frac{2r^{2}\big{(}\,1+\sum_{i=1}^{\infty}a_{i}(r)t^{i}\,\big{)}{\rm
d}z{\rm
d}\bar{z}}{(1+\frac{z\bar{z}}{2})^{2}}\,+\,\big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\big{)}{\rm
d}r^{2}\,\Big{)}$

satisfies the Einstein equations

(3.1)
$\check{E}{}_{ij}=\check{R}_{ij}-\check{\Phi}\check{K}_{i}\check{K}_{j}-\check{\Psi}\check{L}_{i}\check{L}_{j}-(\check{\rho}+\check{p})\check{u}_{i}\check{u}_{j}-\tfrac{1}{2}(\check{\rho}-\check{p})\check{g}_{ij}=0.$

Here $\check{K}_{i}$ and $\check{L}_{i}$ are the null 1-forms corresponding to
the pair of outgoing-ingoing null vector fields

$K=K^{i}\partial_{i}=\partial_{t}{\color[rgb]{0,1,0}+}\Big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\Big{)}^{-\tfrac{1}{2}}\partial_{r}\quad\mathrm{and}\quad
L=L^{i}\partial_{i}=\partial_{t}{\color[rgb]{1,0,0}-}\Big{(}\,1+\sum_{i=1}^{\infty}b_{i}(r)t^{i}\,\Big{)}^{-\tfrac{1}{2}}\partial_{r},$

via $\check{K}_{i}=\check{g}_{ij}K^{j}$ and $\check{L}=\check{g}_{ij}L^{j}$,
and the 1-form vector field $\check{u}_{i}$ corresponds to the future
oriented222Note that now $t<0$ (!) timelike unit vector field

$\check{u}=\check{u}^{i}\partial_{i}=-t^{-1}\partial_{t},$

via $\check{u}_{i}=\check{g}_{ij}\check{u}^{j}$.

Before giving the explicit formulas for the power expansions of functions
$\check{\Phi}$, $\check{\Psi}$, $\check{\rho}$ and $\check{p}$ appearing in
this theorem, we make the following remark.

Remark.

  * •

The Einstein equations (3.1) are equations with an energy momentum tensor
consisting of gravitational radiation propagating with spherical fronts
outward (along $K$) and inward (along $L$); it also consists of a perfect
fluid moving along the present eon’s cosmological time $T=-\int t{\rm d}t$ .
Each front of the spherical wave present in the past eon that reached the
$t=0$ surface in the present eon produces (i) a _spherical outward wave_ ,
going along $K$ out of this sphere with energy density $\check{\Phi}$, (ii) a
_spherical inward wave_ , going along $L$ to the center of this sphere with
energy density $\check{\Psi}$, and (iii) a bit of a _perfect fluid_ with
energy density $\check{\rho}$ and isotropic pressure $\check{p}$.

For the solutions $\nu(t,r)$, $\mu(t,r)$ of the past eon’s Einstein’s
equations (2.3)(2.4), which were given in terms of the power series expansions
as $\nu(t,r)=\sum_{i=3}^{k+2}a_{i}(r)t^{i}+\mathcal{O}(t^{k+3})$ and
$\mu(t,r)=\sum_{i=3}^{k+2}b_{i}(r)t^{i}+\mathcal{O}(t^{k+3})$ in Theorem 1,
the formulae for the power series expansions of the energy densities
$\check{\Phi}$ $\check{\Psi}$, $\check{\rho}$ and the pressure $\check{p}$ are
as follows:

$\displaystyle\check{\Phi}=$
$\displaystyle-\frac{9f}{r^{3}}t^{-3}\,+\,\frac{9f^{\prime}}{r^{3}}t^{-2}\,+\,\frac{1}{2r^{4}}\big{(}8f^{\prime}-11rf^{\prime\prime}\big{)}t\,+\,\frac{3}{4r^{5}}\big{(}5f^{\prime}-5rf^{\prime\prime}+3r^{2}f^{(3)}\big{)}\,+$
$\displaystyle\frac{9}{40r^{6}}\big{(}16f^{\prime}+5ff^{\prime}-16rf^{\prime\prime}+8r^{2}f^{(3)}-3r^{3}f^{(4)}\big{)}t\,+$
$\displaystyle\frac{1}{120r^{7}}\big{(}420f^{\prime}+1068ff^{\prime}-30rf^{\prime}{}^{2}-420rf^{\prime\prime}-384rff^{\prime\prime}+210r^{2}f^{(3)}-70r^{3}f^{(4)}+19r^{4}f^{(5)}\big{)}t^{2}\,+$
$\displaystyle\dots+\mathcal{O}\big{(}t^{k-3}\big{)},$
$\displaystyle\check{\Psi}=$
$\displaystyle-\frac{9f}{r^{3}}t^{-3}\,+\,\frac{6f^{\prime}}{r^{3}}t^{-2}\,+\,\frac{1}{2r^{4}}\big{(}2f^{\prime}-5rf^{\prime\prime}\big{)}t^{-1}\,+\,\frac{3}{4r^{5}}\big{(}f^{\prime}-rf^{\prime\prime}+r^{2}f^{(3)}\big{)}\,+$
$\displaystyle\frac{1}{40r^{6}}\big{(}24f^{\prime}-75ff^{\prime}-24rf^{\prime\prime}+12r^{2}f^{(3)}-7r^{3}f^{(4)}\big{)}t\,+$
$\displaystyle\frac{1}{60r^{7}}\big{(}30f^{\prime}+39ff^{\prime}+75rf^{\prime}{}^{2}-30rf^{\prime\prime}+33rff^{\prime\prime}+15r^{2}f^{(3)}-5r^{3}f^{(4)}+2r^{4}f^{(5)}\big{)}t^{2}\,+$
$\displaystyle\dots+\mathcal{O}\big{(}t^{k-3}\big{)},$
$\displaystyle\check{\rho}=$ $\displaystyle
3t^{-4}+\frac{18f}{r^{3}}t^{-1}\,-\,\frac{18f^{\prime}}{r^{3}}\,+\,\frac{-6f^{\prime}+9rf^{\prime\prime}}{r^{4}}t-\,\frac{3}{4r^{6}}\big{(}9f^{2}+3rf^{\prime}-3r^{2}f^{\prime\prime}+2r^{3}f^{(3)}\big{)}t^{2}\,+$
$\displaystyle\frac{3}{20r^{6}}\big{(}-24f^{\prime}+105ff^{\prime}+24rf^{\prime\prime}-12r^{2}f^{(3)}+5r^{3}f^{(4)}\big{)}t^{3}\,-$
$\displaystyle\frac{1}{20r^{7}}\big{(}60f^{\prime}+96ff^{\prime}+120rf^{\prime}{}^{2}-60rf^{\prime\prime}+72rff^{\prime\prime}+30r^{2}f^{(3)}-10r^{3}f^{(4)}+3r^{4}f^{(5)}\big{)}t^{4}\,+$
$\displaystyle\dots+\mathcal{O}\big{(}t^{k-1}\big{)},$
$\displaystyle\check{p}=$ $\displaystyle
t^{-4}+\frac{6f}{r^{3}}t^{-1}\,+\,\frac{1}{r^{4}}\big{(}2f^{\prime}-rf^{\prime\prime}\big{)}t+\,\frac{1}{2r^{6}}\big{(}18f^{2}+3rf^{\prime}-3r^{2}f^{\prime\prime}+r^{3}f^{(3)}\big{)}t^{2}\,-$
$\displaystyle\frac{3}{20r^{6}}\big{(}-8f^{\prime}+45ff^{\prime}+8rf^{\prime\prime}-4r^{2}f^{(3)}+r^{3}f^{(4)}\big{)}t^{3}\,+$
$\displaystyle\frac{1}{30r^{7}}\big{(}30f^{\prime}+57ff^{\prime}+45rf^{\prime}{}^{2}-30rf^{\prime\prime}+39rff^{\prime\prime}+15r^{2}f^{(3)}-5r^{3}f^{(4)}+r^{4}f^{(5)}\big{)}t^{4}\,+$
$\displaystyle\dots+\mathcal{O}\big{(}t^{k-1}\big{)}.$

In these formulas all the _doted_ terms are explicitly determined in terms of
$f$ and its derivatives (I was lazy, and typed only the terms adapted to the
choice $k=6$ in Theorem 1).

The following remarks are in order:

Remarks.

  * •

Note that since in $\check{M}$ the time $t<0$, the requirement that the energy
densities are positive near the Big Bang hypersurface $t=0$ implies that

$f>0$

in addition to $f^{\prime}>0$, the requirement we got from the past eon.
Indeed, the leading terms in $\check{\Phi}$ and $\check{\Psi}$ are
$\check{\Phi}=\check{\Psi}=-\frac{9f}{r^{3}}t^{-3}$, hence $\check{\Phi}$ and
$\check{\Psi}$ are both positive in the regime $t\to 0^{-}$ provided that
$f>0$. Note also that $f>0$ and $f^{\prime}>0$ are the only conditions needed
for the positivity of energy densities, as the leading term in $\check{\rho}$
is $\check{\rho}\simeq 3t^{-4}$, and is positive regardless of the sign of
$t$.

  * •

Remarkably the leading terms in $\check{\rho}$ and $\check{p}$, i.e. the terms
with negative powers in $t$, are proportional to each other with the numerical
factor _three_. We have

$\check{p}=\tfrac{1}{3}\check{\rho}+\mathcal{O}(t^{0}).$

This means that immediately after the Bang, apart from the matter content of
two spherical ingoing and outgoing waves in the new eon, there is also a
scattered _radiation_ there, described by the perfect fluid with
$\check{p}=\tfrac{1}{3}\check{\rho}$. So what the _Penrose-Tod scenario does
to the new eon out of a single spherical wave in the past eon_ , is it splits
this wave into _three portions of radiation: the two spherical waves_ , one
which is a dumped continuation from the previous eon, the other that is
focusing, _and in addition a lump of scattered radiation described by the
statistical physics_.

## References

  * [1] Fefferman C. and Graham C. R., (2012), ‘The ambient metric’. _Annals of Mathematics Studies_ , 178, Princeton University Press, Princeton, NJ.
  * [2] Penrose R., (2010), ‘Cycles of Time: An Extraordinary New View of the Universe’, Bodley Head,
  * [3] Tod K. P., (2015), ‘The equations of Conformal Cyclic Cosmology’, _Gen. Rel. Grav._ 47,https://doi.org/10.1007/s10714-015-1859-7