Poincare–Einstein approach to Penrose’s conformal cyclic cosmology

We consider two consecutive eons in Penrose’s conformal cyclic cosmology and study how the matter content of the past eon determines the matter content of the present eon by means of the reciprocity hypothesis of Roger Penrose. 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 a pure radiation energy momentum tensor and with a cosmological constant. Using the Poincare-Einstein type of expansion to determine the metric in the past eon, applying the reciprocity hypothesis to get the metric in the present eon, and using the Einstein equations in the present eon to interpret its matter content, we show that the single spherical wave from the previous eon in the new eon splits into three portions of radiation: the two spherical waves, one which is a damped continuation from the previous eon, the other is focusing in the new eon as it encountered a mirror at the Big Bang surface, and in addition a lump of scattered radiation described by the statistical physics.

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Introduction
Our common view of the universe is that it evolves from the initial singularity to its present state, when we observe the presence of the positive cosmological constant [11][12][13]. This has the remarkable consequence that the universe will eventually become asymptotically de Sitter spacetime. This in particular means that the 'end of the universe'-a hypersurface where all the null geodesics will end bounding the spacetime in the future-will be spacelike [9]. Moreover, it will be conformally flat, pretty much the same as the initial boundary of the universe-the Big Bang [17].
This fundamental facts led Roger Penrose to a cosmology theory proposal termed by him conformal cyclic cosmology (CCC). The proposal has a lot of physical motivations, which can be found in [10] and in numerous lectures of Penrose available even in the public media. In this introduction we will only mention its mathematical background, which will be needed to explain our results.
The main feature of CCC is that it states that its universe consists of eons, each being a time oriented spacetime, whose conformal compactifications have spacelike null infinities I . We recall that the future/past null infinity is a boundary I + /I − of spacetime, where all the future/past null geodesics end.
To avoid confusions in understanding CCC we first emphasize that: • CCC says nothing about this what is the physics in a given eon when the physical age of it is normal; normal here means that the eon is neither too young nor too old. CCC tells what is going on when an eon is either about to die, or had just been born. • In particular, CCC does not require that the eons have the same history! It is conformal cyclic cosmology, and not conformal periodic cosmology!
The framework for CCC was recently shaped by Tod, which can be briefly described as follows (see: [18,19], for details): • The universe consists of eons, each being a time oriented spacetime, whose conformal compactifications have spacelike I − and I + . The Weyl tensor of the four-metric on each I is zero. • Eons are ordered, and the conformal compactifications of consecutive eons, say the past one and the present one, are glued together along I + of the past eon, and I − of the present eon. • The manifold M of these glued eons is the universe. The CCC, as formulated by Tod in [18], tells what is mathematical structure of the CCC universe in the neighborhood of matching surfaces of any two consecutive eons. Each such surface is, in Penroses's imaginative language, a wound of the universe. The neighborhood of each wound, consisting of a time portion of the past eon and a time portion of the present eon, is a bandage region of the universe-a (conformal) time sandwich in the universe, in which each wound is bandaged to heal the trauma of the (conformal) transition through the Big Crunch/Big Bang. • Each bandage region is equipped with the following three metrics, which are conformally flat at the wound: * a Lorentzian metric g which is regular everywhere, * a Lorentzian metricǧ, which represents the physical metric of the present eon, and which is singular at the wound, * a Lorentzian metricĝ, which represents the physical metric of the past eon, and which infinitely expands at the wound.
• In a bandage region, the three metrics g,ǧ andĝ, are conformally related on their overlapping domains.
• How to make this relation specific is debatable, but Penrose proposes thať This is called reciprocity hypothesis. • The metricǧ in the present eon is a physical metric there. Likewise, the metricĝ in the past eon is a physical metric there. • Of course, the metricǧ in the present eon, and the metricĝ in the past eon, as physical spacetime metrics, should satisfy Einstein's equations in their spacetimes, respectively.
To answer a natural question on how to make a model of Penrose's bandaged region of two eons, one needs a function Ω, vanishing on some spacelike hypersurface, and a regular Lorentzian four-metric g, such that ifǧ = Ω 2 g satisfies Einstein equations with some physically reasonable energy momentum tensor, thenĝ = 1 Ω 2 g also satisfies Einstein equations with possibly different, but still physically reasonable energy momentum tensor. This is a question similar to the question posed and solved by Brinkman [2]. In 1925 he asked 'when in a conformal class of metrics there could be two nonisometric Einstein metrics?'. Brinkman found all such metrics in dimension four. In every signature.
In CCC the problem is similar. On one hand it seems even simpler: the same function Ω should lead to two conformally related but different solutionsǧ = Ω 2 g andĝ = Ω −2 g of Einstein equations, with a prescribed energy momentum tensor on theM part, and a reasonable energy momentum tensor on the otherM. This creates a highly overdetermined system of PDEs, which may have no solutions at all, or may create algebraic constraints on the unknown variables, resulting in obtaining the solutions in explicit form (see section 2.2 for an example of such a situation). On the other hand, the problem is not so easy, because the three metrics as in (1.1) obtained from the process of solving Einstein equations on both sides of the Big Crunch/Big Bang hypersurface I must be conformally flat on I. This last requirement is automatically satisfied e.g. if one looks for the metrics g,ǧ andĝ which are conformally flat everywhere in the bandage region. It is a reasonable simplifying assumption if one wants to test implications of Penrose's CCC proposal in particular situations, such as in the case of metrics conformal to the Robertson-Walker metrics (see section 2 of the present paper). However to see the implications of the the full CCC setting, the assumption of conformal flatness of the whole bandage region is too strong, and in the general case one is led to consider the full initial value problem on a conformally flat spacelike hypersurface I for the Einstein equations with a particular energy momentum tensor for one of the physical metricsĝ orǧ. This can be in principle done by considering results of Friedrich [4,5] for the conformal data on I applied to the Starobinsky expansion [15] as proposed in the context of CCC by Tod [17]. In the present paper, rather than the Starobinsky expansion we use the Poincare-Einstein type of expansion generalizing for the purpose of CCC the approach to conformal geometry due to Fefferman and Graham [3]. In our section 3, we first formulate this Poincare-Einstein approach to CCC, and then in section 4 we use our new approach to see what's going on with a spherical wave passing from the past eon to the present one if it is obeying these newly established rules of CCC.
In section 2 we describe how to produce conformally flat bandage models of CCC. Throughout the rest of the article, and in particular in section 4 we abandon the conformal flatness everywhere, and create a physically appealing and satisfactory CCC bandage region model which is conformally flat at the Big Bang/Big Crunch hypersurface only.

Conformally flat models in Penrose's conformal cyclic cosmology
To illustrate various CCC bandage region concepts in conformally flat case we will assume in this entire section that all the three bandage metricsĝ, g andǧ are conformal to Friedman-Lemaître-Robertson-Walker metric g test = −dt 2 + Ω 2 (t)r 2 0 dχ 2 + sin 2 χ dθ 2 + sin 2 θ dφ 2 , (2.1) with (locally) spherical spatial sections κ = 1. Before passing to the CCC details we recall the relevant information about this metric and a perfect fluid.

Bandage region in FLRW framework with perfect fluids without cosmological constants
We use this explicit solutions to the Einstein field equation (2.2) to create a conformally flat everywhere bandaged region of two consecutive eons via the Penrose-Tod scenario. On doing this we go back to the Penrose-Tod's bandage triple (ǧ, g,ĝ) and: • We take g as g Einst , g = g Einst ; • We takeǧ = g test = Ω 2 (η)g Einst . This satisfies Einstein's equations with perfect fluid witȟ p =wμ. This means that the conformal scale function Ω = Ω(η) is: • We take asĝ = Ω −2 (η)g Einst .
We have the following theorem relating the polytropes in two consecutive eons: is such thatǧ = Ω 2 g Einst satisfies Einstein's equations, witȟ Λ = 0, and with the energy momentum tensorŤ of a perfect fluid, whose pressurep is proportional to the energy densityμ, viap =wμ,w = const, thenĝ = 1 Ω 2 g Einst satisfies Einstein's equations, withΛ = 0, and with the energy momentum tensorT of a perfect fluid, whose pressurep and the energy densityμ are related byp =ŵμ witĥ The Ricci scalar of the metricĝ iŝ

Remark 2.2.
In CCC the consecutive eons should have spacelike Is. For this the Ricci scalarR of the physical metricĝ must be positive at the wound surface (see [9], p 353, or [18], p 8). This together with the dominant energy condition for the fluid in the past eon, −1 ŵ 1, shows that possible values of theŵ parameter is −1 ŵ < 1/3.
If we have the past eon filled with the perfect fluid withp =ŵμ andŵ ∈ [− 1, 1/3 [, then the reciprocity hypothesis transforms it to a present eon filled with the perfect fluid witȟ w = − 1 3 (2 + 3ŵ). This in particular means that • if the past eon is the de Sitter space,ŵ = −1, then the present eon is filled with radiation, w = 1/3; • if the past eon is filled with a gas of domain walls (see [7], p 219-220), then the present eon is filled with dust,w = 0; • if the past eon is filled with the gas of strings (see [7], p 227), then the present eon is also filled with gas of strings; • This continues: the dust in the past eon is transformed into a gas of domain walls in the present eon.
Going along the intervalw = − 1 3 (2 + 3ŵ) on the (ŵ,w) plane, with −1 ŵ < 1/3, we eventually reach the point (ŵ,w) = (1/3, −1). This point is forbidden however, since for w = 1/3 the I + of the past eon becomes null. To see how the radiationŵ = 1/3 passes through the Bang surface one needs to considerĝ as a solution of Einstein equations with a positive cosmological constantΛ and with the energy momentum tensor of perfect fluid witĥ p = 1/3μ. We will discuss this point in more detail in section 2.3.

Remark 2.3.
The transitions of eons' fluids: cosmological constant → radiation, gas of domain walls ↔ dust and gas of strings ↔ gas of strings was first observed in reference [8]. These transformations appeared there as mysteriously quantized for only five values ofŵ, which were integer multiples of the number 1/3. The above result, which states that all values ofŵ from the interval −1 ŵ < 1/3 are possible, shows that the 'quantization' discussed in [8] was merely obtained as a consequence of the assumptions made in [8], which restricted the search of solutions to only those which were real analytic in the time variable t. As we have shown here, one does not need to look for solutions of the Einstein equation (2.2) with p = wμ in the restricted power series form as in [8]. The Einstein equations solve completely in terms of elementary functions! The general solution is not analytic at t = 0; the solutions which are analytic are those discussed in [8].

Bandage region in FLRW framework with perfect fluids and cosmological constants
To analyze what happens if the conformally flat past eon satisfies Einstein's equations with a cosmological constant and with a radiative perfect fluid we return to the general FLRW metricĝ = Ω(t) −4 −dt 2 + Ω 2 (t)r 2 0 g S 3 , with the original Friedmann time variable t. Then the condition thatĝ satisfies Einstein's equationŝ 2 ,ŵ = const, and the cosmological constantΛ, is equivalent to the following ODE for Ω: We want thatǧ = Ω 4ĝ satisfies the same kind of Einstein's equationš Ric − 1 2Řǧ +Λǧ = (1 +w)μǔ ⊗ǔ +wμǧ, withǔ = −dt, and the cosmological constantΛ, andw = const. This gives another second order condition for Ω, which when compared with (2.4) gives: Differentiating this identity with respect to t, and eliminating the second derivative of Ω by means of the ODE (2.4) we get the next identity relating Ω and Ω . This reads: where to avoid de Sitter solutions on both sides of the bandage region we assumed that Using this assumption again, after an elimination of the unknown Ω from the equations (2.5) and (2.6), we get the following identity which must be satisfied by the unknown constantsΛ, Λ andŵ:ΛΛ Thus, a necessary condition for both Ω and Ω −1 to describe the polytropes, is that either one of the Λs is zero, orŵ is of the 'radiation-Λ' type. The case of two Λs being zero was considered in the previous section; it follows that the casê w = −1 and both Λs nonvanishing leads to the de Sitter space on both sides of the bandaged region. So here we concentrate on the remaining case wheň It follows that in such a case the Einstein equations imply that remarkably alsow = 1/3. This is a generalization of a result from reference [18] stating that if the two Λs are nonvanishing and equal and the past eon is filled radiation type perfect fluid, the present eon is also filled with radiation. More explicitly this case can be integrated to the very end, and we have the following theorem.
Theorem 2.4. The function Ω = Ω(t) given by: has the property that bothǧ = Ω 2 g Einst andĝ = Ω −2 g Einst satisfy Einstein's equations with polytropic perfect fluid equation of state, for whichŵ =w = 1/3 (radiation), and with the corresponding cosmological constantsΛ andΛ. Here g Einst = −Ω −2 dt 2 + r 2 0 g S 3 . This theorem says that incoherent radiation happily passes through the wound. However, cosmological constants can change from any positive value to any other one. In this respect our result is more general than the observation of Tod from [18].

The Poincare-Einstein setting
In reference [18] Tod proposed that in a given bandage region of the universe, the four-metriĉ g of the past eonM should be determined from the initial data, given in terms of the conformal three-metric h 0 on the wound hypersurface, via the Starobinsky expansion [15] Here the past eonM is to be identified with the Cartesian product ofÎ + and the interval [0, [, > 0, and h i , i = 1, 2, 3, . . . , are symmetric rank 2 tensors onÎ + . The tensors h i defining the physical metric of the past eonĝ as in (3.1) are to be determined by the Einstein equations satisfied byĝ inM. Paul Tod further (secretly) requires that the metric formula (3.1) makes sense for τ ∈ ] − , [, with 0 < and that such a metric solves the same Einstein equations for all τ ∈ ] − , [. This produces a metricĝ in the neighborhood M =Î + × ] − , [ of both time sides ofÎ + , i.e. in the full (sufficiently small) bandage region containingÎ + . Once havingĝ there is still a problem in how to defineǧ in M: in order to use the reciprocity hypothesis one needs the choice of either Ω or g. Tod in [18] assumes in addition the constancy of the cosmological constantΛ and finds a PDE for Ω, which can be solved in the cases he considers, giving a unique transition fromM to the new eonM.
In our paper we propose a different algorithm for finding the three metricsĝ, g andǧ defining the geometry in each bandage region. This is based on the following observation (mentioned marginally in [18] by Tod, see the beginning of section 2 in [18]): introduce a new variable t = −e −τ and insert it in the Starobinsky expansion (3.1). This will produce an alternative formula forĝ which is: If suchĝ satisfied the Einstein's equationsRic =Λĝ, this would be the Poincaré-Einstein metric for the conformal class [h 0 ] of three-dimensional metrics defined onÎ + . The power series expansion (3.2), precisely in our variable t, is well known in the theory of conformal invariants [3]. In this theory, a result of Fefferman and Graham [3] guarantees that if the dimension N of the conformal manifold with the conformal structure [h 0 ] represented by a (pseudo)Riemannian metric h 0 is odd, and the expansion h t = ∞ i=0 h i t i contains terms of even powers of t (i.e. if h 2k+1 = 0 for all k = 1, 2, . . . ), then the power series expansion (3.2) is uniquely determined up to infinite order in t by the requirement thatĝ satisfies the Einstein equationsRic(ĝ) = Nĝ also up to infinite order. Actually, Fefferman and Graham have also a theorem that handles the situation with nonzero odd terms in (3.2). Since this is more appropriate for the application in the present paper we quote it here 1 : Fefferman-Graham theorem [3] Let h 0 be a Riemannian metric on a three-dimensional manifoldÎ + and let h be a smooth symmetric tensor onÎ + which is traceless, h i j 0 h i j = 0, and divergence free, ∇ j h ij = 0, w.r.t. h 0 . Then there exists a four-dimensional Lorentzian metriĉ satisfying the Einstein equationŝ These conditions on h t at order 3 in t uniquely determine h t to infinite order at t = 0. Moreover, the solution satisfies h i j 0 (∂ 3 t h t ) |t=0 i j = 0. Our discussion above suggests that Tod's formulation of the geometry of bandage CCC regions in terms ofĝ being in Starobinsky expansion could also be possible in terms of the Poincaré-Einstein expansion (3.2) of Fefferman and Graham. This new formulation should be advantageous since the Fefferman-Graham normal form of the metricĝ distinguishes a particular conformal factor t whose zero defines conformal infinity ofĝ. This conformal factor gives then a natural choice for Ω = t which painlessly solves the problem of finding missing Ω out ofĝ.
We therefore propose to give up with the Starobinsky expansion approach and to formulate the problem of finding the three metrics characterizing bandage regions as follows: • Start with a Riemannian conformal class [h 0 ] on a three-dimensional manifold I . The class is represented by a Riemannian three-dimensional metric h 0 of your choice. • Consider the metriĉ . This is given in the new eonM via the Einstein equations red back, in the Synge's way, from right to left.

Spherical wave transition from the past to the present eon
In this section we assume that in the past eon we have a spherical wave carrying energy densitŷ Φ with speed of light toward its Big Crunch hypersurfaceÎ + . This in particular means that inM we have a given vector field K i , which is null with respect to the metricsĝ =ĝ i j dx i dx j , It also means that the metricĝ satisfies the Einstein equationŝ with this given null vector field K i . We postpone for a while the question how the spherical symmetry of the wave with propagation vector K i is implemented. First we adapt our new procedure of obtaining all the three metricsĝ, g andǧ, to the bandaged region of the above defined matter inM. In this adaptation we will chose the spherical wave inM such that the conformal class [h 0 ] on the Big Crunch hypersurfaceÎ + is flat.

The ansatz and the model for the past eon
We start with a conformal class [h 0 ] represented by the flat three-dimensional metric Then as h t in (3.3) we take the spherically symmetric one-parameter family h t = 2r 2 (1 + ν(t, r)) dz dz 1 + zz where the both unknown functions ν = ν(t, r) and μ = μ(t, r) admit a power series expansion in the variable t such that: ν(0, r) = 0 and μ(0, r) = 0.
This obviously satisfies h t=0 = h 0 and because of our power series assumptions above we have 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 Poincaré-type metricĝ inM: Our past eon manifoldM is therefore parameterized by coordinates (x i ) = (z,z, r, t) with t > 0, r > 0 and z ∈ C ∪ {∞}. Now to implement the spherical symmetry of the wave we consider the following null vector field K onM: Since the postulated propagation vector K of the wave is spherically symmetric, our ansatz for the metricĝ of the wave spacetimeM is now spherically symmetric.

Remark 4.1.
It is worthwhile to note that, regardless if the metricĝ satisfies Einstein's equation (4.1) or not, the vector K is always tangent to a congruence of null geodesics without shear and twist. This represents light rays emanating from the source at the hypersurface r = 0. We require that the Poincaré-Einstein type metric (4.2) satisfies the Einstein equation (4.1) with this null vector field K and some functionsΦ andΛ.
Let us count the unknowns: we do not know the coefficients a i = a i (r) and b i = b i (r) for i = 1, 2, . . . , and we do not know the functionsΦ =Φ(t, r) andΛ =Λ(t, r).

Remark 4.2. Note that if the metric (4.2) satisfies the Einstein equation (4.1) the Einstein tensorĜ
Thus, a priori, the symbolΛ appearing in this equation is a function of all the variables, and in general it is not a constant. The effective energy momentum tensor for the spacetime whose metric satisfies this equation isT i j eff = −Λĝ i j +ΦK i K j , and the contracted Bianchi identity guarantees that∇ only. Of course this does not imply the constancy ofΛ in general case. This is a good feature in a way, because an additional unknown functionΛ prevents the considered Einstein system (4.1) to be overdetermined. Having at our disposal this additional unknown will be particularly useful when we will be solving equation (4.1) by iterating procedure as explained below. On the other hand it would be desirable that the solution we obtain has constantΛ, since in such a caseΛ could be interpreted as the cosmological constant ofM. Also in such a case, the function Φ would has a nice interpretation as the energy density of the spherical wave described by the physical energy momentum tensorT i j =ΦK i K j , which now would be conserved:∇ iT i j = 0. We will see below, that although we do not assume constancy ofΛ, our assumption about spherical symmetry, together with Einstein's equation (4.1), are strong enough to actually guarantee thatΛ is constant up to an infinite order in t.
The strategy of solving the Einstein equation (4.1), for the unknowns a i , b i ,Φ,Λ, and in turn the spherically symmetric wave spacetimeĝ with the wave propagation vector K is as follows: • We first calculate the tensorÊ i j :=R i j −Λĝ i j −ΦK iK j , whereK i =ĝ i j K j , withĝ i j as in (4.2), and K i as in (4.3). This in coordinate basis (x 1 , x 2 , x 3 , x 4 ) = (z,z, r, t), modulo the symmetry, has the following nonvanishing components:Ê 12 ,Ê 33 ,Ê 34 andÊ 44 . We use the Einstein equationsÊ 12 = 0 andÊ 34 = 0 to solve forΛ andΦ. After this move the functionŝ Λ andΦ are explicitly determined in terms of the unknowns μ = μ(t, r), ν = ν(t, r) and their partial derivatives up to order two. Inserting the so obtainedΛ andΦ to the remaining Einstein equationsÊ 33 = 0 andÊ 44 = 0 we obtain a system of two PDEs for the unknowns μ = μ(t, r), ν = ν(t, r) and their partial derivatives up to order two. These two equations in the original coordinates are quite horrible, but introducing a new coordinate T related to t via t = Tr, and denoting the derivative with respect to T as a dot over the function, they can be written in the following form 2 : where F 1 and F 2 are nonlinear differential operators such that F 1 (0, 0) = F 2 (0, 0) = 0 and with the following property: Let s ∈ N. Let μ 0 (T, r) and ν 0 (T, r) be smooth functions such that μ 0 (T, r) = O(T 3 ), ν 0 (T, r) = O(T 3 ) and let μ 1 (T, r) and ν 1 (T, r) be arbitrary smooth functions. Then (4.5) We will return to this property, and the rest of the argument of Graham's letter [6] in section 4.2, when we will comment about the convergence and uniqueness of our solution for the wave. Here we continue in explaining our strategy of solving the remaining two PDEs R 1 = R 2 = 0. • We solve R 1 = R 2 = 0 iteratively, starting with the metric (4.2) with linear in T terms only. Thus, we first assume that the metric is in the form g = 2 (1 +ã 1 (r)T) dz dz The equations R 1 = R 2 = 0 at the order T −1 give thenã 1 =b 1 = 0.
2 I thank Robin Graham for observing that one can write these equations as presented here and explaining this to me in [6].
• Then looking at the metric up to quadratic terms in T we takê g = 2 1 +ã 2 (r)T 2 dzdz Now the equations R 1 = R 2 = 0 starts at the terms of order T 0 . Relating this terms to zero, gives nowã 2 =b 2 = 0. • At the third order in T the situation starts to be interesting: we take noŵ g = 2 1 +ã 3 (r)T 3 dz dz and look at the expressions for R 1 and R 2 . Now, they start at the order T 1 . The equations for R 1 = R 2 = 0 at this order give: This means that to have R 1 = R 2 = 0 up to first order in T the metric must read: where we have introduced an arbitrary function f = f (r) =ã 3 = − 1 2b 3 . • At the fourth order we takê encountering T 2 terms as the lowest order terms in R 1 and R 2 . These terms, when equated to zero give:ã Thus to solve R 1 = R 2 = 0 up to terms of order in T 2 we have to takê • At each order of iteration we can use the obtained μ and ν and its derivatives to calculate the already determinedΛ andΦ. For example, at the 4th order of the metric as above, we calculate thatΛ = 3 + O(t 5 ) and thatΦ = 3 f . Continuing up to infinite order, and using the analysis of our section 4.2, we arrive at the following theorem 3 and (4.5). 3 See also our remark 4.4.

• The metricĝ, or what is the same, the power series expansions ν(t, r)
the Weyl tensor of the solution iŝ In particular, the Weyl tensorŴ i jkl vanishes at t = 0 andΛ > 0 there. With the use of computers we calculated this solution up to the order k = 10, finding explicitly ν = 10 k=3 a k t k and μ = 10 k=3 b k t k . The formulas are compact enough up to k = 8 and up to the order k = 8 they read: as the Poincare-Einstein metric. This is the de Sitter metric withΛ = 3 andΦ = 0. This observation is in accordance with the theory of conformal invariants, which says that in the analytic category the conformal class [h] in dimension 3 determines the Poincare-Einstein metric uniquely. It is worth noticing that in our case of conformally flat [h 0 ], even more general Einstein equation (4.7) than the usual Ric(g) = Λg, imposed on the Poincare-Einstein type expansion also give rise to the unique de Sitter metricĝ, provided that one only admits even powers of t in the expansion. Admitting both, even and odd, powers of t we got an arbitrariness in the metricĝ given in terms of a free function f = f (r). In our solution forĝ the odd power terms in t start at order 3, Note that this is the same order as in the case of odd terms in the power series expansion in the the Fefferman-Graham theorem for the Einstein systemRic(ĝ) = 3ĝ quoted in section 3. Although we have more general Einstein equations imposed on the metricĝ the behavior of odd power solutions is similar to the Fefferman-Graham expansion: the free function f = f (r) in our solution plays the role of traceless divergence free tensor h from Fefferman-Graham theorem. In section 4.2 we show that the appearance of the free function at t 3 is related to the properties of the indicial polynomial matrix of the PDEs we solve to obtain our solution. This polynomial matrix has s = 3 as one of its indicial roots. And it is this root for which the corresponding indicial polynomial matrix has nonzero kernel.
The physical interpretation of the arbitrary function f = f (r) is obvious: it describes the radial modulation of the wave traveling along K i .

Remark 4.4.
We were unable to find a recurrence relation for the functions a i (r) and b i (r) for arbitrary i > 10. We nevertheless claim that such relations do exist and that the corresponding power series are convergent. One reason-Robin Graham's sketch of a rigorous mathematical proof of this fact-is presented in section 4.2. A heuristic reason for these claims is as follows: our solution forĝ is a pure radiation Einstein metric with cosmological constant, which have a shearfree expanding but nonwisting congruence of null geodesics which is tangent to the wave propagation vector K i . All such solutions of Einstein's equations are known. They belong to the Robinson-Trautman class of solutions described e.g. in chapter 28.4 of reference [16]. Our solution is the spherically symmetric solution from this class [20], and can be written in terms of the Robinson-Trautman coordinates [14] as: The trouble is that, because of the appearance of the free function m = m(u) in (4.9), there is no an easy way of getting the explicit coordinate transformation from the Robinson-Trautman null coordinates (ζ,ζ, v, u) to our coordinates (z,z, r, t). Such transformation would bringĝ as in (4.9) to oursĝ from (4.2) in which all the coefficients a i and b i are determined up to infinite order. Anyhow, knowing this transformation or not, the geometric features of our solution with all a i s and b i s determined for i → ∞, show that ourĝ must be identified with the Vaidya solution (4.9). Thus not only the coefficients a i and b i in our solution are determined up to infinite order, but also the power series defining ourĝ converges toĝ given by (4.9).

Remark 4.5.
Note that the last argument above identifying our iterative solution forĝ with the Vaidya's solution is the only justification for the otherwise only 'computer assisted' observation thatΛ of our solution is constant,Λ = 3. The computer calculations to high orders really suggest that it is true to any order, but neither using Fefferman-Graham analysis, nor using the contracted Bianchi identities in some trivial way, show that it is true to infinite order. So the purist would say that the result that for our solution is only a conjecture. We however think that the Vaydia identification argument given in the previous remark is enough to believe thatΛ = 3 exactly.

Sketch of the proof of the convergence of the power series
The following sketch of the proof of the convergence and uniqueness of the solution obtained in the theorem 1 is due to Graham [6]. We invoke the relevant quote from [6] below: '(The Einstein equation (4.4) define) the linear operator It is called the indicial operator. The matrix valued polynomial obtained by replacing T∂ T by s is the indicial polynomial: One has for any smooth vector function v(T, r) = μ 1 (T, r) ν 1 (T, r) . So with μ 0 , ν 0 , μ 1 , ν 1 as (in the first item of our strategy), from (4.4), (4.5) one obtains The solution can now be constructed inductively. Take s = 1 and apply (4.10) with μ 0 = ν 0 = 0. One concludes that μ 1 = ν 1 = O(T) since I(1) is nonsingular. Now take s = 2 and apply (4.10) with μ 0 = ν 0 = 0 to conclude μ 1 = ν 1 = O(T) also for s = 2. However I(3) is singular, so for s = 3 one has exactly the freedom to take Next, take s = 4 with μ 0 ν 0 = T 3 f (r) −2 1 . Since I(4) is nonsingular the order 4 perturbation is uniquely determined by R 1 (μ 0 , ν 0 ) R 2 (μ 0 , ν 0 ) , which vanishes to order 4 by construction. This continues to all orders: the μ 0 ν 0 at step s + 1 is taken to be the previously determined solution at order s, which satisfies R 1 , R 2 = O(T s+1 ) by construction. At each order with s > 3 the solution is uniquely determined since I(s) is nonsingular for all s > 3. Thus there is a unique formal power series solution given the free order 3 coefficient f (r). If f (r) is real analytic, the power series converges. This follows from theorem 2.2 in [1]'. We end up this section with the following corollary which summarizes the results in a long theorem 1.
Corollary 4.6. The Poincaré-Einstein-type metric (4.6) can be interpreted as the ending stage of the evolution of the past eon in Penrose's CCC. The eon has a positive cosmological constantΛ 3, which is filled with a spherically symmetric pure radiation moving along the null congruence generated by the vector field K.

Using reciprocity for the model of the present eon
Now, following the Penrose's reciprocal hypothesis procedure, we summarize the properties of the spacetimeM equipped with the metricǧ obtained fromĝ as in theorem 1, by the reciprocal changeΩ → −Ω −1 = t. In other words, we are now interested in the properties of the metrič g = t 4ĝ . We have the following theorem.
In these formulas all the dotted 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: Remark 4.8.
• Note that since inM the time t < 0, then the requirement that the energy densities are positive near the Big Bang hypersurface t = 0 implies that in addition to f > 0, which was the requirement we got from the past eon. Indeed, the leading terms inΦ andΨ areΦ =Ψ = − 9 f r 3 t −3 , henceΦ andΨ are both positive in the regime t → 0 − provided that f > 0. Note also that f > 0 and f > 0 are the only conditions needed for the positivity of energy densities, as the leading term inρ isρ 3t −4 , and is positive regardless of the sign of t.
• Remarkably the leading terms inρ andp, i.e. the terms with negative powers in t, are proportional to each other with the numerical factor three. We havě This means that immediately after the Bang, apart from the matter content of two spherical in-going and outgoing waves in the new eon, there is also a scattered radiation there, described by the perfect fluid withp = 1 3ρ . • 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 damped continuation from the previous eon, the other that is focusing in the new eon, as it encountered a mirror at the Bang surface, and in addition a lump of scattered radiation described by the statistical physics.

Acknowledgments
This work is inspired by my long term fascination with the approach to differential invariants via curvature invariants of various classes of pseudo-Riemannian conformal structures. This approach is pioneered and developed by Robin Graham. Over the years I had many contacts and discussions with Robin Graham, from which I always learned a lot. Robin was always willing to answer my questions and to give illuminating explanations of his results. Also this paper has a strong imprint of Robin. In particular section 4.2 of the paper including a proof of the convergence of the power series solution from theorem 1 is a quote from a letter from him to me [6]. This puts the main result of the paper on a solid mathematical ground. Since the pandemic canceled Robin's 65th birthday conference, I as an invited speaker for this event, instead of presenting the result prepared for the conference in person, dedicate it to him in this written form.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.