Conformally flat models in Penrose's Conformal Cyclic Cosmology
CCONFORMALLY FLAT MODELS IN PENROSE’S CONFORMALCYCLIC COSMOLOGY
PAWEŁ NUROWSKI
Abstract.
We consider two consecutive conformally flat eons in Penrose’sConformal Cyclic Cosmology and study how the perfect fluid matter contentof the past eon determines the matter content of the present eon by means ofPenrose’s reciprocity hypothesis. Introduction
The C onformal C yclic C osmology or CCC of Roger Penrose [2] is a proposalfor a cosmology which answers the question ‘What was before the Big Bang?’. Themain feature of CCC is that it states that the Universe consists of eons , each being a time oriented spacetime, whose conformal compactifications have spacelike null infinities I .To avoid confusions we first emphasize that: • CCC says nothing about this what is the physics in a given eon when thephysical 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 eitherabout to die, or had just been born . • In particular, CCC does not require that the eons have the same history! Itis Conformal
Cyclic
Cosmology, and not
Conformal
Periodic
Cosmology!The framework for CCC was recently shaped by Paul Tod, and in brief is asfollows (see: [3], for details): • The Universe consists of eons , each being a time oriented spacetime,whose conformal compactifications have spacelike I . The Weyl ten-sor of the 4-metric on each I is zero . • Eons are ordered, and the conformal compactifications of consecutiveeons, say the past one and the present one , are glued together along I + of the past eon , and I − of the present eon . • The vicinity of the matching surface (the wound ) of the past and thepresent eons – this region Penrose calls bandaged region for the two eons– is equipped with the following three metrics , which are conformallyflat at the wound: – a Lorentzian metric g which is regular everywhere, – a Lorentzian metric ˇ g , which represents the physical metric of the present eon , and which is singular at the wound, Date : February 24, 2021.The research was funded from the Norwegian Financial Mechanism 2014-2021 with projectregistration number 2019/34/H/ST1/00636. a r X i v : . [ g r- q c ] F e b PAWEŁ NUROWSKI – a Lorentzian metric ˆ g , which represents the physical metric of the pasteon , and which infinitely expands at the wound. • In a bandage region, the three metrics g , ˇ g and ˆ g , are conformallyrelated on their overlapping domains. • How to make this relation specific is debatable, but Penrose proposes that ˇ g = Ω g , and ˆ g = g , with Ω → on the wound.This is called reciprocity hypothesis . • The metric ˇ g in the present eon is a physical metric there . Likewise,the metric ˆ g in the past eon is a physical metric there . • Of course, the metric ˇ g in the present eon , and the metric ˆ g in thepast eon , as physical spacetime metrics , should satisfy Einstein’sequations in their spacetimes, respectively.To answer a natural question on how to make a model of Penrose’s bandaged regionof two eons, one needs a function Ω , vanishing on some spacelike hypersurface ,and a regular Lorentzian 4-metric g , such that if ˇ g = Ω g satisfies Einsteinequations with some physically reasonable energy momentum tensor, then ˆ g = g also satisfies Einstein equations with possibly different, but still physicallyreasonable energy momentum tensor.This is a question similar to the question posed and solved by H. Brinkman .In 1925 he asked ‘ when in a conformal class of metrics there could be twononisometric Einstein metrics? ’. Brinkman found all such metrics in dimension four . In every signature.In CCC the problem is similar. It seems even simpler: the same function Ω should lead to two conformally related but different solutions ˇ g = Ω g and ˆ g = Ω − g of Einstein equations , with a prescribed energy momentum tensor onthe ˆ M part, and a reasonable energy momentum tensor on the other ˇ M .It seems to be very unlikely that one finds something interesting on ( ˇ M , ˇ g ) , when ˆ T ij and its corresponding ˆ g = Ω − g is given.In this short note, to get some intuitions about the CCC modeling, we check whatwe can do in the conformally flat situation (reasonable, because compatible withthe cosmological principle / FLRW paradigm ), and (various) perfect fluids .2.
Polytropic perfect fluid in FLRW spherical cosmology
In the following we restrict to the FLRW metrics with κ = 1 , g test = − d t + Ω ( t ) r (cid:16) d χ + sin χ (cid:0) d θ + sin θ d φ (cid:1)(cid:17) . It is convenient to introduce a conformal time η = (cid:82) d ta ( t ) so that the FLRWmetric looks g test = Ω ( η ) (cid:16) − d η + r (cid:0) d χ + sin χ (d θ + sin θ d φ ) (cid:1)(cid:17) , i.e. g test = Ω ( η ) g Einst . Here g Einst = − d η + r (cid:0) d χ + sin χ (d θ + sin θ d φ ) (cid:1) is the Einstein Static Universe metric describing the Einstein Static Universe M = R × S , with η being Einstein’s Universe absolute time, and ( χ, θ, φ ) being thespherical angles on S . ONFORMALLY FLAT MODELS IN PENROSE’S CONFORMAL CYCLIC COSMOLOGY 3
This parametrization is very convenient since taking u = − Ω( η )d η , the mostgeneral FLRW metric g satisfying Einstein’s equations (2.1)
Ric − Rg test = ( µ + p ) u ⊗ u + pg test with polytropic equation of state p = wµ , w = const , is given by Ω( η ) = Ω (cid:16) sin w ) η r (cid:17) w if w (cid:54) = − ,and Ω( η ) = Ω exp( bη ) if w = − .3. Bandaged region in FLRW framework with perfect fluids withoutcosmological constants
We use this explicit solutions to the Einstein field equations (2.1) to create a conformally flat everywhere bandaged region of two consecutive eons via thePenrose-Tod scenario. On doing this we go back to the Penrose-Tod’s bandagetriple (ˇ g, g, ˆ g ) and: • We take g as g Einst , g = g Einst ; • We take ˇ g = g test = Ω ( η ) g Einst . This satisfies Einstein’s equations withperfect fluid with ˇ p = w ˇ µ . • We take as ˆ g = Ω − ( η ) g Einst . • Since ˇ g = Ω g satisfying postulated Einstein’s equations has: Ω( η ) = Ω (cid:16) sin (1 + 3 w ) η r (cid:17) w if w (cid:54) = − , and Ω( η ) = Ω exp( bη ) if w = − , then ˆ g = Ω − g satisfies the same kind of Einstein’s equations , but nowwith w replaced by ˆ w such that (1 + 3 ˆ w ) − = − (1 + 3 w ) − , or what is thesame, ˆ w = − / − w . • In other words ˆ g = Ω − g Einst = Ω − ˇ g also satisfies Einstein’s equationswith perfect fluid, but with ˆ p = ˆ w ˆ µ .We have the following theorem relating the polytropes in two consecutive eons: Theorem 3.1. If Ω = Ω( η ) is such that ˇ g = Ω g Einst satisfies Einstein’sequations , with
Λ = 0 , and with the energy momentum tensor ˇ T of a per-fect fluid , whose pressure ˇ p is proportional to the energy density ˇ µ , via ˇ p = ˇ w ˇ µ , ˇ w = const , then ˆ g = 1Ω g Einst satisfies Einstein’s equations , with
Λ = 0 , andwith the energy momentum tensor ˆ T of a perfect fluid , whose pressure ˆ p andthe energy density ˆ µ are related by ˆ p = ˆ w ˆ µ with ˆ w = −
13 (2 + 3 ˇ w ) .The Ricci scalar of the metric ˆ g is ˆ R = − w )Ω r (cid:0) sin w ) η r (cid:1)
1+ ˆ w w if ˆ w (cid:54) = − / and ˆ R = b r )Ω exp(2 bη ) r if ˆ w = − / ,so it is positive if − ≤ ˆ w < / . PAWEŁ NUROWSKI
Remark . In CCC the consecutive eons should have spacelike I s. For thisthe Ricci scalar ˆ R of the physical metric ˆ g must be positive at the wound surface.This together with the dominant energy condition for the fluid in the past eon, − ≤ ˇ w ≤ , shows that possible values of the ˆ w parameter is − ≤ ˆ w < / . If wehave the past eon filled with the perfect fluid with ˆ p = ˆ w ˆ µ and ˆ w ∈ [ − , / , thenthe reciprocity hypothesis transforms it to a present eon filled with the perfect fluidwith ˇ w = − (2 + 3 ˆ w ) . This in particular means that • if the past eon is the deSitter space , ˆ w = − , then the present eon is filledwith radiation , ˇ w = 1 / ; • if the past eon is filled with a gas of domain walls then the present eon isfilled with dust , ˇ w = 0 ; • if the past eon is filled with the gas of strings , then the present eon is alsofilled with gas of strings ; • This continues: the dust in the past eon is transformed into a gas of domainwalls in the present eon .Going along the interval ˇ w = − (2 + 3 ˆ w ) on the ( ˆ w, ˇ w ) plane, with − ≤ ˆ w < / ,we eventually reach the point ( ˆ w, ˇ w ) = (1 / , − . This point is forbidden however,since for ˆ w = 1 / the I + of the past eon becomes null . To see how the radiation ˆ w = 1 / passes through the Bang surface one needs to consider ˆ g as a solutionof Einstein equations with a positive cosmological constant ˆΛ and with the energymomentum tensor of perfect fluid with ˆ p = 1 / µ . We will discuss this point in moredetail in the next section. Remark . The eons’ transitions of fluids (cosmological constant) → (radiation),(gas of domain walls) ↔ (dust) and (gas of strings) ↔ (gas of strings) was observed inRef. [1]. These transformations appeared there as mysteriously quantized for onlyfive values of ˆ w , which were integer multiples of the number / . The above result,which states that all values of ˆ w from the interval − ≤ ˆ w < / are possible,shows that the ‘quantization’ discussed in [1] was merely obtained as a consequenceof the assumptions made in [1], which restricted the search of solutions to onlythose which were real analytic in the time variable t . As we show in this note, onedoes not need to look for solutions of the Einstein equations (2.1) with p = wµ inthe restricted power series form. These equations solve completely in terms of theelementary functions! The general solution is not analytic at t = 0 ; the solutionswhich are analytic are those discussed in [1].4. Bandaged region in FLRW framework with perfect fluids andcosmological constants
To analyse what happens if the past eon satisfies Einstein’s equations with cos-mological constant and radiative perfect fluid we come back to the general FLRWmetric ˆ g = Ω( t ) − (cid:0) − d t +Ω ( t ) r g S (cid:1) . Then the condition that ˆ g satisfies Einstein’sequations ˆ Ric − ˆ R ˆ g + ˆΛˆ g = (1 + ˆ w )ˆ µ ˆ u ⊗ ˆ u + ˆ w ˆ µ ˆ g with ˆ u = − d t/ Ω( t ) , ˆ w = const , and the cosmological constant ˆΛ , is equivalent tothe following ODE for Ω :(4.1) r Ω Ω (cid:48)(cid:48) = (1 + 3 ˆ w )Ω (1 + r Ω (cid:48) ) − (1 + ˆ w )ˆΛ r . ONFORMALLY FLAT MODELS IN PENROSE’S CONFORMAL CYCLIC COSMOLOGY 5
We want that ˇ g = Ω ˆ g satisfies the same kind of Eisntein’s equations ˇ Ric − ˇ R ˇ g + ˇΛˇ g = (1 + ˇ w )ˇ µ ˇ u ⊗ ˇ u + ˇ w ˇ µ ˇ g with ˇ u = − d t , and the cosmological constant ˇΛ , and ˇ w = const . This gives anothersecond order condition for Ω , which when compared with (4.1) gives:(4.2) λ r (1 + ˆ w ) + ˆΛˇΛ r (cid:0) w − w (cid:1) Ω − w ) Ω (1 + r Ω (cid:48) ) + ˇΛ(1 − w ) Ω (1 + r Ω (cid:48) ) = 0 . Differentiating this identity with respect to t , and eliminating the second derivativeof Ω by means of the ODE (4.1) we get the next identity relating Ω and ω (cid:48) . Thisreads:(4.3) r (1 + ˆ w ) − ˆΛˇΛ r (1 + ˆ w )( −
19 + 6 ˆ w + 9 ˆ w )Ω − w ) Ω (1 + r Ω (cid:48) ) + ˇΛ(1 − w ) (7 + 3 ˆ w )Ω (1 + r Ω (cid:48) ) = 0 , where to avoid deSitter solutions on both sides of the bandaged region we assumedthat Ω (cid:48) (cid:54) = 0 . Using this assumption again, after an elimination of the unknown Ω (cid:48) from theequations (4.2) and (4.3), we get the following identity which much be satisfied bythe unkown constants ˆΛ , ˇΛ and ˆ w : ˇΛˆΛ(1 + ˆ w )(1 − w ) = 0 . Thus, a necessary condition for both Ω and Ω − to describe the polytropes, isthat either one of the Λ s is zero, or ˆ w is of the ‘radiation- Λ ’ type.The case of two Λ s being zero was considered in the previous section; it folowsthat the case ˆ w = − and both Λ s nonvanishing leads to the deSitter space on bothsides of the bandaged region. So here we concentrate on the remaining casewhen ˇΛˆΛ (cid:54) = 0 and ˇ w = 1 / . It follows that in such a case the
Einstein equations imply that remarkably also ˇ w = 1 / . This is a generalization of the result of Paul Tod from Ref. [3] statingthat if the two λ s are nonvanishing and equal and the past eon is filled radiationtype perfect fluid, the present eon is also filled with radiation. More explicitly thiscase can be integrated to the very end, and we have the following theorem. Theorem 4.1.
The function
Ω = Ω( t ) given by: Ω = − (cid:113) ˇΛ3 t ) − r √ ˇΛˆΛ sinh(2 (cid:113) ˇΛ3 t )ˇΛ r has the property that both ˇ g = Ω g Einst and ˆ g = Ω − g Einst satisfy Einstein’sequations with polytropic perfect fluid equation of state, for which ˆ w = ˇ w = 1 / (radiation), and with the corresponding cosmological constants ˇΛ and ˆΛ . Here g Einst = − Ω − d t + r g S . This theorem says that incoherent radiation happily passes through thewound . However, cosmological constants can change from any positive value toany other one. This generalizes the observation of Paul Tod from [3].
PAWEŁ NUROWSKI
References [1] Meissner K., Nurowski P., (2017), “Conformal transformations and the beginning of theUniverse”,
Physical Review D , Issue 8, 84016, 1-5;[2] Penrose R., (2010), ‘Cycles of Time: An Extraordinary New View of the Universe’, BodleyHead.[3] Tod K. P., (2015), ‘The equations of Conformal Cyclic Cosmology’, Gen. Rel. Grav. ,https://doi.org/10.1007/s10714-015-1859-7 Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668 Warszawa, Poland
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