Perturbations and the Future Conformal Boundary

Perturbations and the Future Conformal Boundary

A. N. Lasenby,1, 2, ∗ W. J. Handley,1, 2, 3, † D. J. Bartlett,1, 4, 5, 6, ‡ and C.S. Negreanu1, 2, §

1Astrophysics Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge, CB3 0HE, UK2Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK

3Gonville & Caius College, Trinity Street, Cambridge, CB2 1TA, UK4Trinity College, Trinity Street, Cambridge, CB2 1TQ, UK

5Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK6Oriel College, Oriel Square, Oxford, OX1 4EW, UK

(Dated: April 26, 2021)

The concordance model of cosmology predicts a universe which finishes in a finite amount ofconformal time at a future conformal boundary. We show that for particular cases we study, thebackground variables and perturbations may be analytically continued beyond this boundary andthat the “end of the universe” is not necessarily the end of their physical development. Remarkably,these theoretical considerations of the end of the universe might have observable consequences today:perturbation modes consistent with these boundary conditions have a quantised power spectrumwhich may be relevant to features seen in the large scale cosmic microwave background. Mathe-matically these cosmological models may either be interpreted as a palindromic universe mirroredin time, a reflecting boundary condition, or a double cover, but are identical with respect to theirobservational predictions and stand in contrast to the predictions of conformal cyclic cosmologies.

I. INTRODUCTION

Current observations [1, 2] indicate that our uni-verse [3–7] is heading for an ‘asymptotic de Sitter’ state,dominated dynamically by dark energy. An interestingfeature of such a state is that there is a ‘future conformalboundary’ (FCB) present in it. Measured in terms of cos-mic time, this boundary is an infinite time away from us,hence questions about the properties of this boundary,and what happens to various physical quantities whenthey reach it, are perhaps not very pressing, and mayseem academic or abstract at best.

However, in conformal time, which is the elapse ratesuitable for massless particles, the boundary lies only afinite distance away, and will be reached in a fairly shorttime compared to the elapse of conformal time that hasalready occurred since the big bang. Hence for certaintypes of physical quantities, such as perturbations [8–10] in radiation or (massless) neutrinos, the question ofwhat happens to them at the FCB is not academic, butcould even matter in terms of whether the FCB sets anyunexpected boundary conditions on perturbations in theuniverse. More generally, since there certainly are mass-less particles in the universe (photons), it is of interestto consider what happens to perturbations of massiveparticles as well, since these will necessarily be living inthe same universe as the massless particles for which theFCB is rapidly approaching.

These questions have a particular interest within theConformal Cyclic Cosmology of Roger Penrose [11–13],

∗ [email protected]† [email protected]‡ [email protected]§ [email protected]. Current address Microsoft Research,

Cambridge

where the FCB is taken, via an infinite rescaling of thescale factor a, to correspond to the big bang of a further‘cycle’ of the universe. Some of the work to be discussedhere may indeed be relevant to this model, but it turnsout that the type of development beyond the FCB thatseems most natural in the current approach does not cor-respond to a new big bang, but to something differentthat we argue is the obvious analytic extension of thescale factor evolution to that point. Thus issues aboutthe background development of the universe in relation tothe FCB, and not just the evolution of perturbations, aregoing to be relevant to what follows, and form a themeof the paper.

In terms of which perturbations to consider, there isan unfortunate trade-off as regards massless particles be-tween the possibility of getting analytic results on the onehand, and having some degree of realism, on the other.Since analytic solutions are very valuable for guiding in-tuition, we will start with an unrealistic case for whichwe can work out everything analytically, and then moveon to include at least an element of realism. Specifically,we will start by considering a background universe thatis composed of, in stress-energy tensor terms, radiationand a cosmological constant. This is not so unrealisticin itself, in that it is a reasonable approximation to ouractual universe in its early and late stages. The lack ofrealism applies to how we initially treat the perturbationsin such a universe. Radiation in the actual early universeis in close contact with free electrons which scatter it insuch a way as to isotropise the radiation in its rest frame.This results in being able to treat the radiation, includingits perturbations, as a perfect fluid with an equation ofstate parameter, w of 1/3. In this case analytic solutionsare available, and worked out here in Sec. II, for all theperturbed fluid quantities, and we are able to discuss interms of analytic functions what happens to them lateron in their development, as the FCB is approached.

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However, of course in reality, the availability of freeelectrons to scatter off ceases once recombination ispassed, and though the universe is reionised again atlater times, the mean free path of the photons meansthat at all stages after recombination we should be treat-ing photons not as a perfect fluid, but, like neutrinos, viaa distribution function in photon momentum for whichwe develop and solve a Boltzmann hierarchy. This addsconsiderably to the complexity, and all hope of analyticsolutions is lost. We are not interested in very accu-rate calculations here, however, since we are already tak-ing the background evolution as that of just radiationand Λ, with no matter present (the matter necessary toisotropise at early epochs can be assumed to be just traceamounts, with no dynamical effects). Thus in Sec. III wetake an indicative approach, in which we truncate theBoltzmann hierarchy for ` > 2. This enables us to treatthe radiation perturbations (or massless neutrinos per-turbations, which would obey the same equations), asthose in an imperfect fluid in which anisotropic stress isdriven by the velocity perturbations. This gives a mod-icum of realism, whilst allowing analytic power series ex-pansions to be carried out at the FCB, which aid greatlyin understanding what is going on.

Prior to the discussion of cold dark matter (CDM) per-turbations, Sec. IV looks at the behaviour of particles,both massive and massless, near the FCB, and considerstheir geodesic equations, showing that in conformal timemassive particles can be thought of as reflected, whilemassless particles pass straight through. Also consideredare alternative interpretations of the universe beyond theFCB, which may be either thought of as a symmetrical‘palindromic’ evolution of both background and pertur-bations, or as a form of reflecting boundary conditionsgenerated by the double-cover of the first epoch of cos-mic time that conformal time creates.

In Sec. V, we move on to consider perturbations to acold dark matter (CDM) component, and what happensas these approach the FCB. Here, the background uni-verse is taken to be composed of CDM and dark energy,so for this case it is the radiation that is missed out.The benefit is that fully analytic solutions are availablefor all quantities, which again can guide our intuitionin terms of what happens at the FCB and may be ofuse even when considering more realistic scenarios. Thebackground universe solutions for this case are also ofinterest, and may be new as regards their expression inspecial functions.

In what follows we shall present the results for per-turbations and background solutions in an intertwinedmanner, since how the perturbations behave as they ap-proach the FCB is a factor in the arguments for whathappens to the background scale factor evolution afterthe FCB. Also, as regards radiation, we shall first de-scribe in detail the results for the unrealistic case whereit is treated as a perfect fluid throughout. After this,we show how this approach can be repaired, and howthe presence of anisotropic stress leads to some interest-

ing differences with the perfect fluid case. Finally, wediscuss the CDM perturbations, and their analytic prop-erties. A joint analysis of cold dark matter and radiationperturbations, and the effect of the FCB in this case,will be presented in a parallel paper [14], which will alsoconsider some observational consequences which wouldfollow if some of the ideas presented here were taken asapplying to the actual universe.

As a final word of introduction, we want to offer somewords of reassurance to a perhaps sceptical reader, whohaving reached this point, is feeling nervous about theprospect of future boundary conditions being relevant toprocesses which are presumably completely causal in na-ture, and are set in train at the big bang or just after-wards. Of course, this is a very reasonable objection,which stated in this way we completely share. However,the point we are making is that a treatment of pertur-bations needs to consider modes which have periodicityin time not just space, and in a linearised treatment weneed to consider modes which are finite everywhere, bothin space, and crucially, in time. If they were not, thena linear treatment would not be valid. Thus on thesegrounds we feel that in terms of boundary conditions theargument can be made that the behaviour of modes inthe future should be considered as well as their behaviourin the past.

II. PERTURBATIONS IN FLAT-Λ UNIVERSEWITH PERFECT-FLUID RADIATION

For simplicity we will be working throughout in theconformal Newtonian gauge, in which the metric (assum-ing a flat universe) is

ds2 = a2((1 + 2Ψ) dη2 − (1− 2Φ)

(dx2 + dy2 + dz2

)),

(1)where η is conformal time.

This means that we will only be considering scalar per-turbations, but this is enough for our purposes, and themetric in Eq. (1) has the advantage that all the gaugedegrees of freedom in defining scalar perturbations arealready fixed.

We will consider the derivation of the fluid perturba-tion equations for a more realistic radiation component,and how to link with a Boltzmann hierarchy, in Sec. IIIbelow. Here, as explained in the Introduction, we wish toconsider an unrealistic but nevertheless instructive casewhere radiation is treated simply as a perfect fluid withequation of state parameter w = 1/3.

We follow the notation and approach to perturbationsin Liddle & Lyth [9] Chapter 8, and using either theirequations, or the general treatment to be given in Sec. III,we can easily find the perturbation equations appropriateto the w = 1/3 perfect fluid case. For this case it is wellknown (and we discuss again in Sec. III), that the absenceof anisotropic stress means that the ‘potentials’ Ψ and Φare the same. Thus for the remainder of this section onlythe Newtonian potential Φ appears.

3

We can use the constraint equations to solve for thevelocity perturbation V and the density perturbation δ,directly in terms of Φ, and the propagation equation isthen a second order equation in Φ alone. The backgroundquantities are the scale factor a and Hubble parameterH. The equations for the perturbations are then

V =3k(aHΦ + Φ

)2a2(3H2 − Λ)

,

δ = −2(

3H2a2Φ + 3ΦaH + k2Φ)

a2(3H2 − Λ),

(2)

and

Φ + 4aHΦ + 13

(4a2Λ + k2

)Φ = 0, (3)

where an overdot denotes derivative with respect to con-formal time η.

We can accompany these with the equations for thebackground quantities. These are

a = a2H, H = −2

3a(3H2 − Λ

), ρ =

1

8πG

(3H2 − Λ

).

(4)Note these entail the further (background) relations

H = −16πG

3ρ, ρ =

3C

8πGa4, (5)

where C is a constant with dimensions L−2 which wewill discuss shortly. Note that using C we can rewritethe expressions for V and δ as

V =a2k

(aHΦ + Φ

)2C

,

δ = −2a2

(3H2a2Φ + 3ΦaH + k2Φ

)3C

.

(6)

A further first order relation that follows is

δ − 4Φ = −4

3kV. (7)

A. Features of the background solution

The solution we want for the background equationsis one which starts with a big bang, and ends with anasymptotic de Sitter phase. Expressed as a derivativewith respect to cosmic time t, there is a first order equa-tion available in H alone, namely

dH

dt=

2Λ

3− 2H2, (8)

(see Eq. (4) above). To get one when working with con-formal time derivatives, we need to eliminate a, which wedo via the density, obtaining

H = −2

3(3C)1/4

(3H2 − Λ

)3/4. (9)

In this form it is clear that there is a family of big bangsolutions, in which conformal time is scaled proportionalto C−1/4. Simultaneously, from the relation

η =

∫dt

a, (10)

we see that the dimensionless scale factor a is scaled byC1/4. All such solutions are effectively identical — theyjust have a different ‘unit’ for conformal time, which isdimensionful.

We can settle on a convenient value of C to use, inthe following way. First we show that, quite generally,with no assumption about C, equality of the energy den-sities corresponding to radiation and Λ, happens halfwaythrough the conformal time development of the universe.

Then we fix a scale for a, and hence conformal time,by setting a = 1 at this halfway point. This has the con-sequence that there is then a ‘reflection symmetry’ aboutthis halfway point, with the development of the scale fac-tor after it being the reciprocal of the development beforeit.

So we look first at where the energy densities are equal,in terms of conformal time development. The equationfor conformal time in terms of a is

η =

∫da

a2H, (11)

To evaluate this, we need H as a function of a, which iseasy to obtain by eliminating ρ. This gives

H2 =Λ

3+C

a4. (12)

We note that C controls the early universe behaviour,while Λ controls the late universe behaviour. Using thisin Eq. (11) yields

η =

√−i√

3

ΛCF

1

31/4

√i

√Λ

Ca, i

, (13)

where F is the incomplete elliptic integral of the firstkind. This is interesting (and plotted in Fig. 1), but notso useful for our immediate purpose. It is easier to usethe integral for η directly:

η(a) =

√3

Λ

∫ a

0

1√a′4 + 3C

Λ

da′. (14)

Taking the integral all the way to a = ∞ to get thetotal elapse of conformal time in a flat-Λ radiation-onlyuniverse, we get

ηtot =

(ΛC

3

)−1/4 Γ(

14

)Γ(

54

)Γ(

12

) . (15)

Now from the Friedmann Eq. (12), we see that theenergy densities of radiation and the vacuum are equal

4

–3

–2

–1

0

1

2

3

0.5 1 1.5 2 2.5 3

FIG. 1. log10(a) versus conformal time η, measured in units

of 1/√

Λ, in the case where aeq = 1.

at

aeq =

(3C

Λ

)1/4

. (16)

Also, by employing the transformation a 7→ a2eq/a, it

is easy to see that

η(aeq) =

√3

Λ

∫ aeq

0

1√a′4 + 3C

Λ

da′,

=

√3

Λ

∫ ∞aeq

1√a′4 + 3C

Λ

da′ = ηtot − η(aeq).

(17)

Hence, as stated, the energy densities are equal at thepoint halfway through the conformal time developmentof the universe, for any value of C, which means thestatement is true in any flat-Λ pure radiation universe.

A convenient way to fix the scaling of a, and hencealso determine the units of conformal time, is to requireaeq = 1. This makes the development of a symmetricabout the conformal time midpoint. We can see this ex-plicitly in Fig. 1, generated using Eq. (13), which showslog10(a) versus η in this case. Choosing a different (con-stant) scaling for a would just move this plot up and downby a constant amount, and stretch the horizontal axis bya constant factor. With aeq = 1 we see that the curve fora is antisymmetric under reflection both left-right andtop-bottom, with these symmetries corresponding to thetransformations a 7→ 1/a and η 7→ ηtot − η respectively.(Further discussion of these transformation and their re-lation to inversion symmetry of the Friedmann equationcan be found in [15].)

The value of C which gives this behaviour is C = Λ/3,and for this case

H2 =Λ

3

(1 +

1

a4

), (18)

which exhibits a neat symmetry between early and lateepochs.

We note that the total conformal time elapsed for thiscase is

ηtot =

√3

Λ

Γ(

14

)Γ(

54

)Γ(

12

) ≈ 3.21135

√1

Λ, (19)

so that η inherits its units of time from 1/√

Λ.Finally, we need to consider the conversion to cosmic

time t. This is given by

t =

∫da

aH. (20)

We note that the absolute scale of a cancels out of thisexpression, so the units of t are determined simply bythe units of the physical parameter H. Thus, unlike thecase with conformal time, there is not an extra scalingto be fixed. Carrying out the integral for a general Cand writing H∞ =

√Λ/3 for the value of the Hubble

parameter at infinite cosmic time, we obtain

t =1

2H∞sinh−1

(H∞a

2

√C

), (21)

which becomes

t =1

2H∞sinh−1

(a2), (22)

if the above value of C is used.

B. Solution for the Newtonian potential Φ

A way of solving for the development for the Newto-nian potential Φ, is to transform Eq. (3), in which botha and H appear, and the derivatives are with respect toconformal time, into an equation in which only a appearsand the derivatives are taken with respect to a. The com-plicated dependence of a on η implied by Eq. (13) is thenavoided for the purposes of solution.

We also make a further variable dimensionless by writ-ing the comoving wavenumber k as k = K

√Λ. These

changes lead to

a(1 +a4)Φ′′+ 2(3a4 + 2

)Φ′+a

(4a2 +K2

)Φ = 0, (23)

where a prime denotes derivatives with respect to a.This can be solved in terms of a Heun function

Φ(a) = (1 + a4)1/4 exp(

12 i tan−1(a2)

)×

HeunG

(−1,

1

4(5− iK2), 1,

5

2,

5

2,

1

2, a2i

),

(24)

where the particular combination of solutions which leadsto this form has been chosen so that Φ is real and tendsto 1 as a → 0. Indeed, the series for Φ at small a givenby this expression is

Φ(a) ≈ 1− K2

10a2 +

(K4

280− 1

7

)a4 − . . . (25)

5

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2

a

FIG. 2. Plot of the Newtonian potential Φ(a) as a functionof scale factor a for normalised wavenumber K = 10.

A plot of the Heun function expression in Eq. (24) forK = 10 is shown in Fig. 2. The plot is generated byMaple, which, however, only seems to be able to eval-uate the function correctly out to a ≈ 2.5 for this case.Due to this, plus the ability to calculate some asymptoticvalues which we will need below, it is useful to develop analternative representation for Φ. A useful new expressionwas found to be

Φ(a) =3√

1 + a2K2 + a4

a3K√K4 − 4

sin(K√K4 − 4ψ(a)

), (26)

where ψ(a) is the integral

ψ(a) =

∫ a

0

a′2√1 + a′4 (1 + a′2K2 + a′4)

da′. (27)

(See [16] for some information on integral representationsof Heun functions.) We can see straight away from thisintegral expression that for small a, ψ(a) will behave likea3 and hence from Eq. (26) Φ(a) will behave like a sincfunction, which we can indeed see in Fig. 2.

The integral in Eq. (27) can be evaluated analytically,and we find

ψ(a) = e−iπ4

(Π(eiπ4 a, 1

2 i(K2 −

√K4 − 4

), i)

−Π(eiπ4 a, 1

2 i(K2 +

√K4 − 4

), i))

,(28)

where Π is the incomplete elliptic integral of the thirdkind, and we are using Maple’s notation for the orderand meaning of the arguments. Inserting this ψ(a) intoEq. (26) then gives a fully analytic result for Φ(a), andone that is more convenient to use in practice than theHeun result in Eq. (24). This is so since (a) Maple canevaluate this function over the whole range of a withouterrors; (b) asymptotic expressions are available as a →∞; and (c) derivatives with respect to a can be taken andthe results found are still in terms of elliptic functions,meaning that the resulting expressions for the derivedquantities δ and V are also analytic.

C. Initial conditions for the perturbations

Above we have been working with a form of solutionfor the Φ equation in a that tends to a constant (chosenas 1) as a→ 0. We now explore the initial conditions forΦ(a) more generally to understand how the chosen formarises.

The Φ equation in a in the form we are using (with the

choice of Λ/3 for C and k = K√

Λ) is given by Eq. (23).Using the following trial form for Φ

Φ = aν(c0 + c1a+ c2a

2 + c3a3 + . . .

), (29)

we find that ν is constrained to either 0 or −3, and thatthe accompanying series are

Φ(a) = c0

(1− K2

10a2 +

(K4

280− 1

7

)a4 − . . .

), (30)

or

Φ(a) =d0

a3

(1 +

K2

2a2 −

(K4

8− 1

2

)a4 + . . .

), (31)

respectively. Since Eq. (23) is a linear and second order,the solutions of which these are the first terms of span theentire space of solutions for Φ(a) — all other solutionsare linear combinations of these.

Now, self-evidently, Eq. (31) blows up as the origin isapproached, hence it is not admissible as a solution forour current setup. The reason for this, is not because ofits singularity per-se — for example several other quan-tities which we think are physical, such as the density orHubble parameter, blow up as the origin is approached— but because we have used linearised equations for theperturbations, having Φ blow up means that the condi-tions for linearisation are not fulfilled. Hence we need torestrict to Eq. (30) as the only possible linear mode.

To clarify this point further, we examine the behaviourof δ and V as the origin is approached. Expressed interms of Φ(a), we find the following general expressionsfor these as a function of a

δ = −2(1 + a2K2 + a4

)Φ− 2a

(1 + a4

)Φ′, (32)

and

V =

√3

2aK√

1 + a4 (Φ + aΦ′) . (33)

These lead to the following series at the origin

δnon-singular = c0

(−2− 7K2

5a2 +

(23K4

140− 4

7

)a4 + . . .

),

(34)

δsingular =d0

a3

(4− 2K2a2 −

(K4

2− 2

)a4 + . . .

),

(35)

V non-singular =

√3Kc02

(a− 3K2

10a2 + . . .

), (36)

6

–6

–4

–2

0

2

4

6de

lta, P

hi a

nd V

0.5 1 1.5 2 2.5 3eta

FIG. 3. Evolution of perturbation quantities in conformaltime for K ≈ 9.58. Φ is shown in red, δ in black and V inblue.

V singular = −√

3Kd0

(1

a2− K4

8a2 + . . .

). (37)

δ = δρ/ρ is a pure number which needs to be � 1in order for the linearisation to be valid, and the valueof V corresponds to the modulus of the (assumed non-relativistic) velocity perturbation v/c, and so again hasto be small. Thus the singular solution is not possiblehere (as already said for Φ, which is also dimensionless).Note, however, the density itself is tending to infinityas a → 0 is approached, so since δ tends to a constant(−2c0), the actual density perturbation, δρ, is infiniteeven in the non-singular case.

The fact that we eliminate one of the two possiblemodes via this argument is part of the reason it was pos-sible to predict the sequence of peaks in the CMB powerspectrum [17, 18], even before the theory of inflation wasavailable. Effectively ‘starting from rest’, which is whatinflation achieves via the ‘coming over the horizon’ recipe,will be the same as using only a non-singular solution, ifthe cosmic epoch at which a given perturbation comesover the horizon is sufficiently early.

D. General features of the results

To illustrate the general physical features of the resultsfor the perturbation quantities, we start with a specificcase as illustration. In Fig. 3 we show curves for Φ, δand V plotted as a function of conformal time for a nor-malised wavenumber K near 10. We see that while Φdecays away, δ and V soon settle down into very regular-looking sine waves in conformal time, with no sign ofdecaying away. The velocity V is 90◦ out of phase withthe density perturbation δ, as we expect from Eq. (7).

In Fig. 4 we show the evolution of δ and V again, butthis time with respect to cosmic time t. This is interestingin that it appears to show the quantities ‘freezing out’,and if plotted to higher t (which of course goes to +∞),

–6

–4

–2

0

2

4

6

delta

and

V

2 4 6 8t

FIG. 4. Evolution of δ and V in cosmic time for K ≈ 9.58. δis shown in black and V in blue.

they would not depart visibly from the values alreadyreached by about t = 10 here. This is because we cansee already in Fig. 3 what values they will reach, sincethis latter plot is for the full span of conformal time,which occurs as we saw from Eq. (19) at η ≈ 3.21135

√Λ,

corresponding to the right hand end of the plot.The nature of this type of ‘freezing out’ will be dis-

cussed further below in Sec. IV A, in the context of thebehaviour of matter versus photon geodesics at the fu-ture conformal boundary (FCB). However, the main thingwhich strikes one from the plot against conformal time inFig. 3, and we wish to draw attention to here, is that itis clear that the perturbations in δ and V are marchingtowards the right in a very regular fashion, and show nosigns that they are ‘noticing’ the boundary at η = ηtot.This naturally raises the question of could the pertur-bations pass ‘through’ the FCB, and in this case, whatspace is it they emerge in?

E. The Future Conformal Boundary (FCB)

We now investigate in more detail what happens toboth the perturbations and background solution as theFCB is approached.

To explain the background solution properly, we needto introduce a version of General Relativity in which thesign of the scale factor has significance, and can be mon-itored. In standard GR based solely on the metric, it isonly a2 that has significance in the metric, as can be seenfrom Eq. (1).

This can be done using a tetrad approach to GR (seee.g. [19]), but we are going to indicate the needed re-lationship schematically here, using some notation from‘Gauge Theory Gravity’ (GTG — see [20]), which is par-ticularly convenient for conformal metrics. The notationis for a vector-valued function of vectors, h, which is es-sentially the square root of the metric, and arises as thelocal gauge field corresponding to gauging translations.

7

If the vector it is operating on is b, then the h-functionfor the background solution we are using has the simpleexpression

h(b) =1

a(η)b, (38)

so that (for this case), the ‘translation gauge field’ cor-responds to just a scaling of input vectors by 1/a(η). Atthe FCB, the scale factor a(η) becomes infinite. Thissuggests that a more sensible quantity in which to ex-press the h-function near this point is the reciprocal ofa, which we call s, so s(η) ≡ 1/a(η), and now

h(b) = s(η) b, (39)

which one might argue is a more natural way to expressthe conformal scaling in any case.

The background Eqs. (4) and (5) in the new variables = 1/a are

3s2 = 8πGρ+ Λ, and 3s2− 2ss = −8πGρ

3+ Λ. (40)

If we make the same choices as above w.l.o.g, and bringin the constant C, take this as Λ/3 as before, and work inunits of conformal time such that Λ = 1, we can rewritethese equations as

3s2 = 1 + s4, and 3s2 − 2ss = 1− 13s

4 (41)

It is easy to verify that the second of these equations iscompatible with the derivative of the first.

Taking the difference between the two equations en-ables us to get an equation the solution of which is freeof potential square roots, namely

s = 23s

3, (42)

which is remarkably simple.Again, in order to get things in the simplest form, we

note that η does not appear explicitly in either equationand therefore we are free to shift the origin of η to wher-ever we wish, and we now take this as happening at theFCB itself, so that η = 0 there.

Eq. (42) can be solved in terms of a single Jacobi snfunction but this has a complex argument and imaginaryparameter. An interesting expression in terms of realJacobi elliptic functions which one can find, and whichsolves the first order equation as well, is

s(η) = ±sn(η√3, 1

2

)cn(η√3, 1

2

) dn(η√3, 1

2

). (43)

This appears to be the ‘elliptic’ version of ‘tan’, thoughwe have not seen it called as such. It involves the ‘sin’over ‘cos’ ratio, but then corrected by the dn function. Ithas the same property as ‘tan’ that if we describe its‘range’ by the distance in η between where it is zero

–8

–6

–4

–2

0

2

4

6

8

–3 –2 –1 1 2 3

t

FIG. 5. Plot of the reciprocal scale factor s(η) versus η asgiven by the function in Eq. (43).

and where it becomes infinite, then its values in thesecond half of the range are the reciprocals of thoseat the reflected position (about the midpoint) in thefirst half of the range. (For ‘tan’, this is the identitytan(π4 + θ) = cot(π4 − θ).) This also means it reaches 1at the midpoint. We can now understand the propertieswe have been looking at so far for the scale factor a(η)versus η in this pure radiation flat-Λ universe, as arisingfrom this elliptical ‘tan’ function.

Fig. 5 shows a plot of this function about η = 0 andwhere we have chosen the negative pre-factor in Eq. (43).The idea here is that the portion before η (the locationof the FCB) corresponds to the current epoch, whichhas a positive and decreasing reciprocal scale factor. Wecan see the elliptical tan function smoothly extends thisthrough η = 0, into a universe that is now antisymmet-ric (in η) about the line η = 0 (which remember nowcorresponds to the FCB).

Where s(η) goes to plus infinity as η → −ηtot as givenin Eq. (19), corresponds to the ‘big bang’, and where s(η)goes to negative infinity as η → +ηtot, must correspondto a reflection symmetric big bang in the future.

Interpreting a universe in which the conformal scalefactor (as described in h-function terms) is negative, ischallenging, but it is not clear that there could be anyfundamental objections to it. At the level of the metric,the flip from s to −s is invisible, and the universe tothe right of η = 0 in Fig. 5 looks like a ‘regular’ bigbang model but playing out backwards in time. However,whether the space and time parity inversions implied bya negative factor s in

h(b) = s(η) b, (44)

mean that the universe might be ‘seen’ as playing out for-wards in time is moot, since we cannot have any sensibleform of observer in a radiation-only universe.

What we can say is that we seem to find an unambigu-ous result for how this type of universe extends through

8

0

2

4

6

8

–3 –2 –1 1 2 3

t

FIG. 6. Plot of the reciprocal scale factor s(η) versus η ac-cording to the Penrose Conformal Cyclic Cosmology proposal.

the FCB, and it is not the extension which Penrose sug-gests in his ‘Conformal Cyclic Cosmology’ proposal [11–13]. This involves an infinite rescaling of the scale factorat the FCB, so that what succeeds the decreasing seg-ment of the l.h.s. of the plot in Fig. 5, is a repetitionforwards in conformal time, rather than backwards inconformal time, of the same segment. In other words,the Penrose proposal is for an evolution of the reciprocalscale factor of the kind shown in Fig. 6, for which thereis no evidence in terms of the equations presented here.

F. Evolution of perturbations through the FCB

We now consider how the radiation perturbations be-have as the FCB is approached. This is facilitated by theremarkable fact that, just like the background equations,the perturbation equation can be put in a form that isinvariant under a 7→ 1/a, and so we can use the solutionswe have already found when expanding out of the bigbang to find solutions valid when expanding about theFCB.

The form in which the equation for the Newtonian po-tential is invariant is where we work not with Φ(a) butϕ(a) ≡ a2Φ(a). Then Eq. (23) above for Φ in terms of abecomes

a2(1 + a4)d2

da2ϕ+ 2a5 d

daϕ+

(a2K2 − 2a4 − 2

)ϕ = 0.

(45)If we now make the change of independent variable a 7→s = 1/a, then the equation becomes

s2(1+s4)d2

ds2ϕ+2s5 d

dsϕ+(s2K2 − 2s4 − 2

)ϕ = 0, (46)

so we see that indeed the equation is invariant underusing the reciprocal of the scale factor.

This means that we do not have to do any additionalwork to find the form of the Φ solutions near the FCB.

We can directly use the series for Φ coming out of thebig bang already found in Eqs. (30) and (31). The onlychange we need to make is to replace a by s and thenmultiply by s4, corresponding to the fact that it is a2Φthat is invariant, not Φ. This results in the series

Φ(s) = c0s4

(1− K2

10s2 +

(K4

280− 1

7

)s4 − . . .

), (47)

and

Φ(s) = d0s

(1 +

K2

2s2 −

(K4

8− 1

2

)s4 + . . .

). (48)

The dramatic thing here of course, is that both of theseseries are non-singular at s = 0. Thus the FCB does notset any further constraints, by itself, on the developmentof the perturbations in Φ. Any ‘incoming’ Φ(s) has twodegrees of freedom (Φ and Φ′), and will be able to matchto a linear combination of these two solutions. Of coursewe also need to look at δ and V . The equations forthese are not invariant under a 7→ s = 1/a, and so wemust do some further work to find the power series forthese. Expressed as a function of Φ(s) (i.e. the analogueof Eq. (32)), we find

δ =2

s4

(s(1 + s4

)Φ−

(1 + s4 +K2s2

) dds

Φ

). (49)

From Eqs. (47) and (48), the lowest power contained inΦ(s) in the neighbourhood of the FCB is s1, and it isnot immediately evident this will be enough to make δnon-singular, due to the overall 1/s4 factor in Eq. (49).However, inserting Eqs. (47) and (48) into Eq. (49), somecancellations occur, and we obtain

δ(s) = c0

(6− 3K2s2 +K2

(4 +

K4

4

)s4 + . . .

), (50)

and

δ(s) = d0s

(4− 2K4 +K2

(1 +

K4

4

)s2

+(4−K4

)s4 + . . .

),

(51)

respectively, which are both non-singular at s = 0. Per-haps surprisingly, we see that the Φ series with the lowestleading power in s leads to the δ series with the highestleading power and vice versa.

We can do the same for the velocity perturbation V ,obtaining

V =

√3

2s3K√

1 + s4 (Φ− sΦ′) , (52)

as the analogue to Eq. (33), and

V (s) = −√

3c02

s

(3K − K3

2s2 +K

(1

2+K4

40,

)s4 + . . .

)(53)

9

Phi

0

10

20

30

40

50

1 2 3 4 5 6

FIG. 7. Evolution of perturbation quantities in conformaltime for the case shown in Fig. 3 (K ≈ 9.58), extendedthrough the Future Conformal Boundary to the full rangeof η.

and

V (s) = −√

3d0

2

(K3 +K

(2− K4

2

)s2 + +

K3

2s4 + . . .

),

(54)as the two series.

Since all of the series are non-singular, this explainshow δ and V are able to approach the FCB ‘withoutnoticing’ — there is no singularity there, therefore theyare able to march straight through regardless of theirphase and magnitude. This is at a certain level com-forting, since if one of the available two modes at theFCB had been singular, as at the big bang, then thiswould have set an unexpected boundary condition, re-sulting presumably in a restriction on the set of K valueswhich could be used.

However, since we are able to continue both the back-ground and perturbations unambiguously through theFCB, we can ask of the perturbations: what happensto them to the right of the boundary? Do they continuebeing non-singular as they advance into this space? Sincethe FCB is in fact a completely regular point of the sys-tem of equations, the issue of whether a given mode isallowable presumably needs to be settled by looking atwhat happens as the perturbations approach the nextgenuine singularity. This occurs as the density starts be-coming infinite again, as we approach the next (thoughtime reversed) big bang, which occurs at the right of thediagram in Fig. 5.

Fig. 7 shows what happens if we extend the integrationof the case shown in Fig. 3 through the FCB and towardsthe next big bang. We see that δ and V continue tooscillate regularly until we start approaching the rightboundary, where clearly they and Φ start diverging.

What is happening, of course, is that they are failingto join onto the regular series given earlier for small a(e.g. in Eq. (30) for Φ). By symmetry these must still

0

2

4

6

8

n P

i/2

2 3 4 5 6K

FIG. 8. Plot of the function ϑ(K) defined in Eq. (55). Thisfunction being equal to nπ/2 for positive integer n definesthe range of allowable values for the (normlised) comov-ing wavenumber K. The first few values of nπ/2 and thewavenumbers K they thereby define, are also shown.

be valid at the right end of Fig. 3 where |a| is becomingsmall. Clearly the functions δ, V and Φ, as they approachη = 2ηtot contain a non-zero component of the singularseries (e.g. Eq. (31) for Φ, and this means they diverge.

Now, as argued earlier, it is simply not possible to usemode functions which diverge when carrying out lineari-sation. Thus, if we believe the region to the right ofthe FCB has some validity, then the case just discussedis not allowable, and we are indeed faced with a non-trivial boundary condition, but we have found it entersat η = 2ηtot, not η = ηtot.

The required boundary condition comes about fromthe fact we need Φ, δ and V to be either symmetric oranti-symmetric about the FCB if they are to remain non-singular as the right hand boundary is approached. Thiswill mean they are recapitulating the modes in whichthey left the first big bang, which we argued above hadto be non-singular. Put differently, we can now realisethat boundary conditions are set at the two end pointsof the range (first and second big bangs), and these willforce a discrete set of K to be used; those in which asuitable number of cycles complete over the range.

We can derive this constraint on K analytically, byconsidering the expression for Φ which we achieved inEqs. (26) and (28). It is not hard to show this leads tothe requirement

nπ

2= ϑ(K) ≡

√K2 + 2

K2 − 2K

(EllipticK

(1√2

)−EllipticPi

(1

2− K2

4,

1√2

)), n = 1, 2, 3, . . .

(55)

where we are using the Maple notation for the ellipticintegrals, since otherwise there is a bad clash betweendifferent types of K!

We plot in Fig. 8 the function defined on the r.h.s.

10

n K1 2.183129712952 3.116688651353 4.018623475954 4.90240252065

TABLE I. The first few allowed K-values.

of Eq. (55), which we have called ϑ(K). One can see

that it starts from 0 at K =√

2, and then settles downfairly rapidly to a linear form. We may express boththe initial and large K behaviour in terms of the ηtot

parameter defined in Eq. (19), i.e. the elapse of conformaltime between the first big bang and the FCB, which mayalso be written as ηtot =

√3 EllipticK(1/

√2) in units of

conformal time where Λ = 1.We find

ϑ(K) ≈21/4√

3(2η2

tot − 3π)

6ηtot,

√K −

√2, (56)

for small K >√

2 and

ϑ(K) ≈ ηtot√3K, (57)

for large K. As already stated, the function fairly rapidlysettles down to this latter form, and so the intervals be-tween allowable K solutions soon become regular, andthese occur at an interval which tends rapidly to

∆K ≈√

3π

2ηtot≈ 0.847. (58)

The K spectrum defined by our two boundary condi-tions is therefore discretised, but fairly regularly spaced,except for the first few values. These are given explicitlyin Tab. I.

It will be observed that we are not allowing K =√

2 asa possibility, i.e. the solution for n = 0. From Eqs. (26)

and (28) we see that K =√

2 is effectively a ‘boundary’between propagating and non-propagating solutions andit is interesting in itself that we get a lower limit on Kvalues even without appealing to future boundary con-ditions. However, if we go ahead and integrate the per-turbations using this value for K we find that it is notsymmetric (or anti-symmetric) about the midpoint, andwe obtain the results shown in Fig. 9 indicating clearlythat this case violates the boundary conditions.

In Fig. 10 we show the results for the smallest success-ful value of K, K ≈ 2.1831 . . ., as given in Tab. I. We seethat for this case Φ and δ are antisymmetric about themidpoint, and V is symmetric.

In Fig. 11 we also show the results for the second suc-cessful mode, K ≈ 3.1167. This time Φ and δ are sym-metric about the midpoint, and V is antisymmetric. Thisbehaviour alternates as one might expect as one movesup through the modes.

Phi

–10

–8

–6

–4

–2

0

2

4

delta

, Phi

and

V

1 2 3 4 5 6eta

FIG. 9. Evolution of Φ, δ and V over the full range in ηfor modes with K =

√2, which forms the boundary between

propagating and non-propagating waves.

Phi

–6

–4

–2

0

2

4

6

delta

, Phi

and

V

1 2 3 4 5 6eta

FIG. 10. Evolution of Φ, δ and V over the full range in η formodes with the first value of K that successfully meets theboundary conditions at each end, K ≈ 2.1831.

Phi

–4

–2

0

2

4

6

delta

, Phi

and

V

1 2 3 4 5 6eta

FIG. 11. Same as for Fig. 10, but for the second mode, K ≈3.1167.

11

–6

–4

–2

0

2

4

6de

lta, a

^2 P

hi a

nd V

1 2 3 4 5 6eta

FIG. 12. Evolution of ϕ, δ and V over the full range in η forthe 10th mode, K ≈ 10.0765. (ϕ = a2Φ is shown multipliedby 100 to bring it on the same scale as the others.) ϕ is shownin red, δ in black and V in blue.

Finally, for completeness, we show in Fig. 12 a case forhigher K, in which K has the value for the 10th mode,K ≈ 10.0765. We can now reveal that the case we usedseveral times earlier, K ≈ 9.58, was in fact chosen ascorresponding to the 10th mode minus one half, so as togive a good example of something which fails to satisfythe boundary conditions, as shown in Fig. 7. In Fig. 12,instead of plotting Φ, we plot the function ϕ(a) = a2Φ(a),which we showed earlier in Eq. (46) is invariant under thereciprocity transformation a 7→ s = 1/a. It is interestingthat it is this version of Φ that has the same characteras δ and V , of propagating with effectively unchangedamplitude over the whole range of η. If we had picked anodd mode, however, ϕ would have diverged at the mid-point. We can understand this in terms of the Eqs. (47)and (48), which express the behaviour of Φ in terms ofs at the midpoint. The first of these applies when Φ iseven (such as for the 10th mode), and since it goes likes4, is still non-singular after division by s4 to form ϕ(s).When Φ is odd, only the Eq. (48) is selected, and thisthen behaves like s−1 for ϕ and diverges (since Φ ∝ s+. . .for this case). The divergence of ϕ near s = 0 for oddmodes is not a problem since it is only Φ that we need toremain small compared to 1 to satisfy the linearisationconditions.

As a final point to discuss in this w = 1/3 perfect fluidcase, we could put off for the moment the fact that oursetup does not correspond even approximately to physi-cal reality (as discussed in the Introduction) to ask whatwould be the observational consequences if the aboveanalysis needed to be taken seriously, and the K spec-trum is indeed discrete?

The first obvious consequence is that there would beno primordial fluctuation power below a value of K givenby the first entry in Tab. I, K ≈ 2.1831, which we labelKmin. We defined K via k = K

√Λ, and the currently

observed value of Λ ≈ 1.21 × 10−7 Mpc−2, so Kmin cor-

responds to a comoving wavenumber of

kmin ≈ 7.6× 10−4 Mpc−1. (59)

This is quite an interesting number in connection withindications for a decrease in power in the primordialpower spectrum at large angular scales, which happens ataround this point in k. However, since our perfect fluidis not a possible physical description of the radiation af-ter it has lost the frequent interaction with electrons bythe end of recombination, we now need to look at a morerealistic setup, and see if any of the above effects andconsiderations come into play there.

III. DERIVATIONS AND RESULTS FOR AMORE REALISTIC RADIATION COMPONENT

We now consider radiation perturbations and their be-haviour as they approach the future conformal boundaryfor the case where the radiation component is not treatedas a perfect fluid. We will do this via a Boltzmann hier-archy approach, but where we truncate the hierarchy atharmonics ` > 2. This enables us to work in terms of afluid still, but now an imperfect one, with an anisotropicstress that is driven by the velocity perturbations. Thiswill enable us to have at least an indication of the be-haviour in a more realistic case, where the radiation isdecoupled from the matter.

We may follow the treatment in Chapter 8 and 11 of[9], except some of the expressions there are for the zero Λcase, and also some relevant equations contain misprints.Thus we give the full set of equations we need here, andpoint out the differences from [9] where appropriate. Weshall give the equations first in a first order propagationplus constraint format, and then consider other versionslater. Since there is now anisotropic stress, the potentialΨ now features as well Φ in the quantities to be propa-gated. We note that since the Boltzmann evolution equa-tions used are specific to the radiation case, we give allresults for w = 1/3 only.

First of all, the relevant definition of the anisotropicstress is

Π =3k2 (Φ−Ψ)

a2 (3H2 − Λ). (60)

If we set Λ = 0 and compare with the result (8.36) in [9],we see this has the opposite sign. It is not quite clearwhy this happens, since the signs associated with it inthe Boltzmann hierarchy in Section 11 of [9] agree withwhat we find. In any case we believe Eq. (60) is correctfor our current purposes.

The expressions

Θ0 =δ

4, Θ1 =

V

3, Θ2 =

Π

12, (61)

relate the spherical harmonic modes Θi used in the Boltz-mann hierarchy equations, to the fluid quantities already

12

defined (see (11.10) in [9]). The Boltzmann hierarchy canthen be written

δ = −4

3kV + 4Φ,

V = k

(δ

4− 1

6Π + Ψ

),

Π =12k

5

(2

3V − 3Θ3

).

(62)

In the third equation, we have brought in an ‘unin-terpreted’ fourth Boltzmann mode, Θ3, which it is notpossible to relate to fluid quantities. However, this iswhere truncation of the series for ` > 2 comes in. Wedeclare that for our purposes this term can be ignored,and so can ‘close’ the Boltzmann series via the relation

Π =8k

5V. (63)

Comparing our results so far with those in [9], we notethat in their Section 11.2, they say that neutrinos, if oneignores their mass, should satisfy the same Boltzmannhierarchy equations as photons when there are no colli-sions, which is what we are assuming here. However, inthe case where the Θ3 mode is put to zero, they obtain

Πν =4k

5Vν , (64)

i.e. a coefficient of 4/5 instead of the 8/5, we have justfound in Eq. (63). We think that this is likely a misprint.

Similarly, although [9] agree with our Eq. (62) for V asapplied to the neutrino case (their (11.16)), they insteadgive

Vγ = k

(1

4− 2Πγ + Ψ

), (65)

as applied to the radiation case (their equation (11.28) inthe case there is no optical depth due to matter). Again,we think this just corresponds to misprints, and that theequation for V in Eq. (62) is correct.

Continuing now to give the first order propagationequations for quantities, we find:

Ψ = aH (2Φ− 3Ψ) +2

15ka2V

(3H2 − Λ

),

Φ = −aHΨ +2

3ka2V

(3H2 − Λ

).

(66)

Meanwhile, the non-derivative constraint equation is

a2(3H2 − Λ

)(4aHV + kδ) + 2k3Φ = 0. (67)

If we differentiate this with respect to conformal time,and then use the above derivative relations, we obtaina multiple of the constraint equation itself, showingthe propagation equations and constraint are consistent.This is not a full test of the relations, however, since nei-ther the Ψ potential or Π appear in the constraint. We

can, however, go through all the Einstein equations andBianchi identities for the system explicitly, and one findsthat everything is properly satisfied given the above re-lations, so we take them as a valid starting point for theperturbation analysis.

Note that if we wish to return to the previous perfectfluid case, then this is equivalent to truncating the Boltz-mann hierarchy at ` > 1, which means Π is set to 0 andwe ignore the Π equation in Eq. (62). The only otherchange necessary in the equations already given, is in theΨ equation in Eq. (66), which becomes

Ψ = aH (2Φ− 3Ψ) +2

3ka2V

(3H2 − Λ

)= −aHΦ +

2

3ka2V

(3H2 − Λ

),

(68)

in agreement with the Φ equation, since of course Ψ = Φin this case. We mention this to note that the Ψ equationdoes not go smoothly to the perfect fluid case when weswitch off the anisotropic stress, as we can see from thedifferent factors in front of the a2V

(3H2 − Λ

)terms in

Eqs. (66) and (68), i.e. 2/(15k) and 2/(3k) respectively.This is a feature of truncating the Boltzmann hierarchyat different points.

Note a result which will be useful below is if we attemptto find an equation in Φ alone, which can parallel thesecond order equation for Φ given in the perfect fluid casein Eq. (3) and for which it was possible to find analyticsolutions.

We find

2 aH(−60 a2k2Λ + 2040H2a4Λ + 45 k4

−800 Λ2a4 + 648 k2H2a2)

Φ

+(45 k4 + 8568H4a4 − 720 Λ2a4

+918 k2H2a2 − 1416H2a4Λ)

Φ

+30 aH(−68 a2Λ + 15 k2 + 168 a2H2

)Φ

+15(5 k2 − 20 a2Λ + 42 a2H2

) ...Φ = 0.

(69)

which indeed parallels of Eq. (3) in the perfect fluid case,but is third rather than second order, and unlike theperfect fluid case appears not to have analytic solutions.We will still find it useful shortly, however, in findingthe behaviour of series solutions at the big bang and theFCB.

A. Some series and numerical solutions

As an initial step in seeing what changes in behaviourthe anisotropic stress causes, we consider the evolutionout of the big bang. The third order Eq. (69) just foundis useful for this, since given that we know a in terms ofH, via

a = −3

2

H

3H2 − Λ, (70)

13

then we just need to set up series for H and Φ and weare thus dealing with the complete problem, since Ψ, δand V can all be derived from Φ (see below).

The analogue of this for the perfect fluid case has al-ready been discussed in detail in Sec. II C, where weshowed how various possible solutions had to be elimi-nated on the grounds that otherwise the linearisation stepis invalidated. The same happens here, and the survivingsolution for Φ is now (written in terms of conformal time,rather than the a used in Sec. II C):

Φ = c0

(1− K2Λ

21η2 +

17Λ2(3K4 − 40

)54936

η4 − . . .

).

(71)The accompanying series for Ψ is

Ψ = c0

(5

7− K2Λ

42η2 +

Λ2(3K4 − 40

)7848

η4 − . . .

), (72)

and hence it is impossible to start the universe off withzero anisotropic stress in this case — Ψ is forced to bedifferent from Φ.

Note, however, that at the start of the universe evo-lution, the radiation component can be assumed to be aperfect fluid, hence we should start as before with zeroanisotropic stress, and only later move over to the regimewith non-zero Π. Given that we have first order equa-tions for all quantities (i.e. for Φ, Ψ, δ and V ), this canbe done easily by just taking as the starting values forfurther evolution, the values reached by the quantitiesat the point where matter/radiation decoupling wouldhave progressed sufficiently (if we were including bary-onic matter as a fluid as well), that the radiation wasno longer being isotropised in the matter rest frame. Inother words, at that point we move to the equations givenin this section for further evolution, having to that pointused the equations of Sec. II instead.

To make these ideas concrete, we now give some exam-ple numerical evolution curves for a case treated in thisway, and where the value of conformal time η at whichthe transition to the new equations takes place is chosento give clear illustrations, rather than being ‘realistic’ -we will attempt something closer to the latter towardsthe end of this section.

In Figs. 13 to 15 we show the evolution of the per-turbed quantities for the case K = 10 where the integra-tion starts from the values reached for these quantitiesat η = 1 (chosen just for illustrative purposes). It isthen continued through the future conformal boundaryusing the series expansions we will derive below, and thenallowed to carry on its evolution numerically in the re-gion beyond the FCB. The figures are plotted in terms ofs = 1/a and hence the evolution in conformal time is infact from right to left. Thus, for example, in Fig. 15, theanisotropic stress evolution starts at s ≈ 1.713, which iswhere the reciprocal scale factor has reached at confor-mal time η = 1 after the big bang, with a value for Πof zero, since it is assumed that up to this point there

FIG. 13. Plot of density perturbation δ (black) and velocityperturbation V (blue) as a function of reciprocal scale factors = 1/a for normalised wave number K = 10 in the ‘Boltz-mann hierarchy’ model for radiation perturbations. The con-formal time evolution starts at the right and progresses left,passing through the FCB at s = 0.

FIG. 14. Same as Fig. 13, but for the potentials Φ (red) andΨ (green).

FIG. 15. Same as Fig. 13, but for the anisotropic stress Π.

14

FIG. 16. Plot of density perturbation δ (black) and velocityperturbation V (blue) as a function of reciprocal scale factors = 1/a for normalised wave number K ≈ 2.605 in the ‘Boltz-mann hierarchy’ model for radiation perturbations. The con-formal time evolution starts at the right and progresses left,passing through the FCB at s = 0. This value of K appearsto correspond to the first allowed mode.

FIG. 17. Same as Fig. 16, but for the potentials Φ (red) andΨ (green).

is some matter available to isotropise the radiation in itsrest frame. Thereafter the evolution is leftwards towardsthe FCB at s = 0 and then into the region beyond theFCB with s < 0.

We have made use of series at the FCB which haveyet to be derived, but the clear implication of these plotsis that the perturbed quantities ‘do not see’ the FCB,but just go straight through it, in the same way as forthe treatment in Sec. II. (The potentials Φ and Ψ bothgo to zero at the FCB, and in this sense ‘see’ it, butas we show below from the series, this does not set anyconstraints). Thus the issues as regards valid values of Kwill be the same as before, i.e. we will want to chooseK sothat the evolution after the FCB does not subsequently‘blow up’, and we expect a similar conclusion in whichwe are forced to use a solution which is either symmetricor antisymmetric (in the relevant quantities) at the FCBitself.

We show in Figs. 16 to 18 the evolution of perturba-

FIG. 18. Same as Fig. 16, but for the anisotropic stress Π.

tions for what appears to be the first ‘symmetric’ mode,which occurs at K ≈ 2.605. (‘Symmetric’ is in quotessince in fact the V perturbation is antisymmetric in thiscase.) The potentials retrace their values before the FCBin their development afterwards, and hence can convergeto the same value at an η of -1, making Π zero there,and hence (assuming the universe now has enough hotmatter for the isotropisation to take over again), the de-velopment can now be the usual perfect fluid one fromthis point onwards towards the symmetric ‘big bang’.

Having established some examples, we now discuss theseries solutions at the FCB, which enable the numericalintegrations to be continued past this point.

A convenient way of establishing behaviours, is to usethe third order equation for Φ, given in Eq. (69), and torewrite this just as a function of s = 1/a, which we cando by eliminating H

5s3(1 + s4)(14s4 + 5K2s2 − 6)d3

ds3Φ

−10s2(62s4 + 15K2s2 + 14s8 − 12)d2

ds2Φ

+s(156K2s6 + 306K2s2 + 45K4s4

−240 + 1492s4 + 252s8)d

dsΦ

−312K2s2 − 1360s4 − 90K4s4 + 240− 432K2s6 = 0.(73)

We can now substitute a series of the form

Φ = sν(c0 + c1s+ c2s

2 + c3s3 + . . .

), (74)

for Φ in order to to discover the ‘indicial indices’ ν thatare possible for Φ at the FCB. This yields the indicialindex equation

(ν − 1)(ν − 2)(ν − 4) = 0, (75)

in other words, ν is restricted to be 1, 2 or 4. The ν = 1and ν = 4 results match the leading order behaviour wefound previously (Eqs. (47) and (48)) in the perfect fluidcase, but we now have ν = 2 as an extra possibility,

15

corresponding to the extra degree of freedom opened upby the equation for Φ now being third order, rather thansecond. We can see that in all three cases Φ is constrainedto be 0 at the FCB, but that since we still have 3 d.o.f.there is no extra constraint coming from this, as alreadymentioned above.

These results translate through to the other potentialΨ, via an expression we can find for Ψ in terms of Φalone. This expression is

Ψ =1

14s4 + 5K2s2 − 6

(5s2(1 + s4)Φ′′

−2s(3s4 + 8)Φ′ + 10(1 + s4 +K2s2)).

(76)

Using this, and substituting the same series as abovefor Φ, we find that the series for Ψ has first term

Ψ = −sν c06

(10− 21ν + 5ν2) + . . . (77)

and hence for the above values of ν starts with the samepower of s as Φ.

We can go further and ask about the value of theanisotropic stress Π at the FCB. Translating Eq. (60)in terms of s, we find

Π =3(Φ−Ψ)K2

s2. (78)

Using Eqs. (74) and (76) this yields the values

Π(s) =

−3K2c1 + . . . for ν = 1

−3K2c0 + . . . for ν = 2

6K2c0s2 + . . . for ν = 4,

(79)

so we can see that despite the s2 in the denominator ofEq. (78), Π is non-singular at the FCB, in all cases.

We can put things into a useful form by rewriting theseries for Φ as

Φ = s(c0 + c1s+ c2s

2 + c3s3 + . . .

), (80)

where now c0 controls the ν = 1 series, c1 controls theν = 2 series and c3 controls the ν = 4 series. (Anycontribution from c2 must be associated with the secondor third term of a ν = 2 or ν = 1 series.)

With c0, c1 and c3 controlling the degrees of freedom,we now find the following series expansions for the re-maining perturbations at the FCB:

V = −1

2K3√

3c0 −5

2K√

3c3s+1

20Kc0

(9K4 − 20

)√3s2

+1

4

√3K

(3K2c3 − 4c1

)s3 + . . .

δ =(−2K2c1 + 10c3

)− 2c0

(K4 − 2

)s+

(−5K2c3 + 4c1

)s2

+1

15K2c0

(9K4 − 14

)s3 +

(4c3 +

3

4K4c3 −K2c1

)s4 + . . .

Ψ = c0s+ 2c1s2 − 7

10K2c0s

3 − c3s4 +1

120c0(9K4 − 20

)s5 + . . .

(81)

These results are what we need in order to understandthe symmetry properties at the FCB. Looking at whatwe have called the first ‘symmetric’ solution above, whichwas shown in Figs. 16 to 18, we can see that this mustcorrespond to c0 = 0, which makes Φ, Ψ and δ symmet-ric, and V antisymmetric. In this process both c1 andc2 remain free. In contrast solutions with the oppositesymmetry require both c1 and c3 to vanish, with only c0allowed to be non-zero. A solution of this type is shownin the previous perfect fluid case in Fig. 10, in which V issymmetric, and δ and Φ (= Ψ for this previous case) areantisymmetric. This was the first mode available in theperfect fluid case, and we might expect the first mode tohave that symmetry here. However, this is prevented bywhat we have just seen about the number of degrees offreedom available in this ‘antisymmetric’ case, as we nowexplain.

The four functions Φ, Ψ, δ and V intrinsically have3 degrees of freedom available, due to the constraintEq. (67). This matches the 3 d.o.f. available at the FCBfor Φ alone, when it is treated via its third order equation(73). In order to be able to reach a point after the FCBwhere Ψ = Φ again, and such that we can continue evolu-tion towards a symmetric big bang in such a way that allperturbations remain non-singular, we are going to need3 d.o.f. If we use a ‘symmetric’ solution at the FCB, wehave seen this fixes c0, leaving 2 d.o.f. available in c1 andc3. The remaining d.o.f. needed is of course provided byK, which we adjust to get the desired overall evolution,thereby discovering its eigenvalues.

On the other hand, if we take the ‘antisymmetric’ caseat the FCB, we have only 1 d.o.f. left, corresponding tothe value of c0, and this combined with the freedom inK is not enough to give us solutions — one more d.o.f.would be needed for us to be able to advance towards afuture big bang in which the perturbations remain non-singular.

This contrasts with the perfect fluid case, where onlytwo d.o.f. are needed for the functions in general, sincethe equation for Φ alone is only second order, and in boththe symmetric and antisymmetric cases at the FCB wehave 1 d.o.f. left there (corresponding to the ν = 1 andν = 4 solutions of Eqs. (47) and (48)). This is then joinedby K to make up the total.

Thus on these grounds, we would predict that in theimperfect fluid case now being treated, we will be ableto successfully find symmetric solutions (in Φ, Ψ andδ, but V will be antisymmetric), but not antisymmetricsolutions.

This indeed appears to be the case numerically.Searching through the K values to find a mode that doesnot ‘blow up’ after the FCB, we find a first successfulvalue at K ≈ 2.605, as already shown, but this effec-tively corresponds to what would have been the secondmode of the perfect fluid case. There is no analogue ofthe first mode of the latter.

Similarly, the next successful K value we find, is K ≈3.89, and this behaves similarly to the 4th mode of the

16

FIG. 19. Plot of anisotropic stress as a function of reciprocalscale factor s = 1/a for normalised wave number K ≈ 3.89 inthe ‘Boltzmann hierarchy’ model for radiation perturbations.This value of K appears to correspond to the second allowedmode.

perfect fluid case. For interest, the plot for anisotropicstress Π in this case is shown in Fig. 19.

We thus believe that only ‘even’ modes can be contin-ued through the FCB without becoming singular. It isof interest to evaluate the K values of these modes formore ‘realistic’ values of the conformal time at which theisotropisation of the radiation finishes than we have so farused, just in case this is of some relevance to the actualuniverse. So far we have just used perfect fluid evolutionup to an arbitrary value of conformal time η = 1, but wecan improve on this as follows.

By putting together the expression for ρ in Eq. (5) withour value of C = Λ/3 and the fact that the energy densityof the radiation is aSBT

4rad, where aSB is the reduced

Stefan-Boltzmann constant, it is possible to show thatthe radiation temperature satisfies

T 4rad =

Λc2s4

aSB8πG, (82)

where s is the inverse scale factor.Putting in the numbers, using an estimate of the actual

cosmological constant Λ from current observations, yields

Trad ≈ 30sKelvin. (83)

Thus a sensible definition of the conformal time atwhich to make the transition from perfect to imperfectfluid might be the conformal time that corresponds to thes which makes Trad = 4000 K. This corresponds approx-imately to the temperature at which decoupling takesplace in our actual universe. We thus need s = 4000/30and using our various expressions above for s in terms ofη, we find this occurs at a conformal time of η ≈ 0.013.This contrasts with the η for switch-over we have beenusing in the examples until now, which was at η = 1. Wecould have carried out the previous examples using the

n K for ηstart = 1 K for ηstart = 0.0131 2.605 2.4768452 3.89 3.7196727

TABLE II. The first two allowed K-values in the cases withimperfect fluid evolution occurring beyond η = ηstart, for thetwo values of ηstart considered in the text.

FIG. 20. Plot of the potentials Φ (red) and Ψ (green) forthe first valid mode (K ≈ 2.476845) in the case where theimperfect fluid development is started at a more realistic valueof conformal time. See text for further details.

new value of η = 0.013 instead, but this more realisticvalue leads to variations of the perturbed quantities withs that have a high dynamic range near the FCB, and arethus unsuitable for illustrating general features.

The important aspect of the new η starting value, ofcourse, is the effect that it has upon the allowed K values.We have only computed the first two such values, andthese turn out to be decreased somewhat compared tothe starting η = 1 case. We show the new values obtainedin comparison with the previous ones, in Tab. II.

As an example of the actual evolution with the newstarting η, we show the development of the potentials Φand Ψ for ηstart = 0.013 and K ≈ 2.476845, which webelieve corresponds to the first allowed mode, in Fig. 20.The plot is as usual in terms of s = 1/a on the x-axis,so reads from right to left as the universe proceeds from‘decoupling’ at the right hand edge, through to the sym-metrical point at the left hand edge. As mentioned, thereare large variations near the FCB itself (s = 0), whichis why we chose ηstart = 1 for the majority of the exam-ples. Despite the very different appearance, the structureof the variations in perturbed quantities is qualitativelysimilar to that found before for K ≈ 2.605, making usconfident that this is indeed the first allowed mode.

17

IV. GEODESICS AND INTERPRETATIONS OFTHE SYMMETRY CONDITIONS

Before moving on to consider the behaviour of colddark matter (CDM) perturbations at the FCB, we willfirst get clear on the behaviour of particles, both massiveand massless, near the FCB, as a guide to what we mightexpect in the way of differences from the radiation case.Additionally we will use these results to briefly discusstwo different interpretations of the nature of the space‘after’ the FCB.

A. Geodesics at the FCB

We aim to look at the behaviour of geodesics as theyapproach the FCB, and in particular whether they can gothrough it. We will also discuss the issue of geodesic com-pleteness, which is a topic of interest for cyclic cosmolo-gies [21]. Note that in a similar way as we used GaugeTheory Gravity (GTG) [20] notation in Sec. II E to dis-cuss evolution of the background quantities through theFCB, we will do the same here. This has the advantage ofaccess to a space of covariant vectors, such as the particleor photon momentum and velocity, which is not readilyavailable in GR, even when using a tetrad approach, andis sensitive to issues about the sign of quantities. As be-fore the results will be discussed in a way such that thereader can treat the notation schematically, rather thanneeding to understand detailed definitions. In addition,in Sec. IV A 3 we give a fully ‘GR-only’ derivation of theimportant result we will find here, that in contrast tophotons, the motion of a material particle as a functionof conformal time, ‘turns around’ at the FCB and headsbackwards in spatial terms, rather than passing throughthe FCB. Note that in this section, we will temporarilyre-use the symbol Φ for parametrising the particle andphoton momentum vectors, which since the Newtonianpotential Φ does not appear here, should not cause anyconfusion.

1. Photons

We start with photons, which are the simpler case. Weparameterise the photon 4-momentum as

p = Φ(λ)(et + er), (84)

where λ is the affine parameter along the path, et and erunit vectors in the time and radial directions respectively,and we are assuming purely radial motion. In this case(radial motion), there is a conserved quantity, p·gr = pr,where pr is the downstairs radial momentum (in a GR

sense) and gr = h−1(er).1 This means that the quantity

E = a(η)Φ, (85)

which we can interpret as the downstairs component ofphoton energy, in a GR sense, is constant. This is justa statement about photon redshift of course, but we arebeing clear here, since it is related to the similar casefor massive particles, that where the constancy comesfrom is the conservation of the (GR) radial momentumcomponent, which E happens to equal.

The geodesic equations for a radially outgoing photonare found to be

Φ′ = − a

a2Φ2 = −HΦ2, η′ =

Φ

a, r′ =

Φ

a, (86)

where the dash temporarily indicates a derivative withrespect to affine parameter λ, and the overdot still indi-cates derivative w.r.t. conformal time η.

Note that

a′Φ + aΦ′ = aη′Φ + aΦ′ = aΦ2

a− a

aΦ2 = 0, (87)

confirms that Eq. (85) is compatible with Eq. (86).Since we are interested in what happens close to the

FCB, we solve these equations in the case where we ap-proximate H as constant, writing

H ≈ H∞ =

√Λ

3. (88)

We also temporarily take the origin of conformal timeas being at the FCB itself, so that the period before theFCB has negative η. With these assumptions, we findthe solution

s = 1/a = −H∞η, (89)

for the inverse scale factor in terms of η, and

Φ =1

H∞λ, η = − 1

H2∞Eλ

, (90)

for the geodesic parameters in terms of λ, where we haveused Eq. (85) to provide a constant in the solution forη. (It is worth noting that the dimensions of all thesequantities work out provided the affine parameter λ hasdimensions L2, which is correct for a photon.)

Since s tends to 0 at the FCB, and becomes negativethe other side, this means that λ has to tend to infinity asthe FCB is approached, and then jump to minus infinityon the other side. This is the same behaviour as forthe scale factor. Meanwhile Φ goes down through 0 andbecomes negative the other side, obeying

Φ = −H∞Eη = Es. (91)

1 Here h−1(b) for a vector b is the ‘inverse adjoint’ of the functionh(b) introduced in Sec. II E.

18

In this sense we have a negative energy photon after theFCB.

For the radial position, r, its derivative is the same asfor η, and so if we denote by r1 the value of r when theFCB is reached, we can write

r = r1 −1

H2∞Eλ

= r1 + η. (92)

The photon, in terms of r position, thus goes smoothlythrough the FCB as a function of conformal time, whichof course it has to, as a massless particle. This is atthe same time as the affine parameter along the photon’spath is first reaching plus infinity, then jumping to minusinfinity at the boundary.

Now the definition of geodesic completeness is that theaffine parameter should be able to take an unrestrictedset of values along the particle geodesic. We see thaton this basis the photon geodesics are complete both asthey approach the FCB and as they move away from it.This is due to the fact that λ is able to reach all the wayto plus or minus infinity. (Note we do not consider theother ends of each worldline, which will have restrictionson λ due to either the big bang or a mirror version in thefuture, and hence not be complete at these boundaries.)

We mentioned just now that photons after the FCBappear to have negative energy, since the locally ob-served energy for an observer moving with the cosmicfluid would normally be taken as p·et = Φ, where et isthe unit length timelike frame vector of such an observer,and we have seen that Φ definitely changes sign after theboundary.

However, for this argument we need to understandwhat happens to observers themselves, and so need toconsider the geodesic equations of massive particle, towhich we now turn.

2. Massive particles

Here we parameterise the particle 4-momentum as

p = m (cosh(Φ(τ))et + sinh(Φ(τ))er) , (93)

where Φ is now the rapidity parameter along the path,which we parameterise with particle proper time, τ , andm is the particle mass.

We are assuming purely radial motion, and again thedownstairs radial momentum, pr = p·gr, will be con-served for this motion, meaning

P = ma(η) sinh(Φ), (94)

will be constant.The geodesic equations for a radially outgoing massive

particle, are found to be

Φ′ = −H sinh Φ, η′ =cosh Φ

a, r′ =

sinh Φ

a, (95)

where the dash now indicates a derivative with respectto the proper time τ .

Taking the derivative of Eq. (94), we find

a′ sinh Φ + a cosh ΦΦ′ = aη′ sinh Φ + a cosh ΦΦ′

= acosh Φ sinh Φ

a− aH cosh Φ sinh Φ = 0,

(96)

confirming Eq. (94) is compatible with Eq. (95).Again, since we are interested in what happens close

to the FCB, we solve these equations in the case wherewe approximate H ≈ H∞. We can express the results asfollows. Writing

X(τ) =H∞

2(τ − τ0) + coth−1

(eΦ0), (97)

we find

Φ = ln (cothX) , a =P sinh(2X)

m

η = − m

PH∞ sinh(2X), r = r1 +

m (1− coth(2X))

PH∞,

(98)for the background and geodesic parameters in terms ofτ . Here we have assumed Φ > 0 and that Φ goes throughthe value Φ0 at τ = τ0, while r reaches r1 at the FCB,where τ →∞.

We note that the ordinary velocity of the particle, v =tanh Φ can be written in terms of X as

v =dr

dη= tanh Φ =

1

cosh(2X), (99)

meaning that the particle is coming to a halt as it ap-proaches the FCB, where X → ∞. We would again saythat the geodesic is complete since the affine parameter,the proper time τ here, extends out all the way to +∞ inreaching the boundary. However, since there is an expo-nential singularity (ether positive or negative) in the var-ious quantities when expressed in terms of proper time,as the FCB is approached, it is not clear what shouldhappen beyond it.

We here note what we have already alluded to earlier,in the discussion of the contrast in pictures of the evo-lution of the radiation perturbations as seen in Figs. 3and 4, which were plotted in conformal time then cos-mic time respectively. Cosmic time is of course just theproper time of a particle at rest with respect to the Hub-ble flow, which the massive particle we are consideringhere approximates to better and better as the FCB isapproached. We saw earlier from these figures that cos-mic time is an unfortunate coordinate for considering theperturbation evolution, since it rapidly ‘freezes out’ andceases to give information, whereas conformal time givesa clear picture of the whole development. This leads usto regard conformal time as the driving independent vari-able in the evolution of quantities, and we now show itleads to a clearer picture not just for radiation pertur-bations (where this could have been expected) but formassive particle geodesics as well.

19

FIG. 21. Plot of the radial coordinate r versus conformaltime for a massive particle initially moving radial outwards.It reaches the Future Conformal Boundary at an r of r1 = 1and thereafter starts moving inwards.

So we now give solutions for the geodesic parametersin terms of conformal time, rather than proper time. Weobtain

Φ = − sinh−1

(PH∞η

m

), s = −H∞η

r = r1 +m (1− cosh Φ)

PH∞= r1 +

m

PH∞−

√η2 +

m2

P 2H2∞.

(100)In terms of this parameterisation, it is quite clear that

nothing is singular at the FCB, and that as η evolvesfrom negative before the FCB, through zero at the FCB,to positive afterwards, everything behaves smoothly. Thesurprise, however, is that we now see that the radial coor-dinate r turns around at the FCB and evolves backwardsafter it. A schematic example of this is shown in Fig. 21.This is in contrast to the photon case, where we hadr = r1 + η, i.e. it just continued incrementing in the waynecessary for an outgoing photon.

We can understand why this behaviour is necessary ata simple level by looking at the contrasting parametri-sations that are necessary for the particle momentum inthe photon and massive particles cases. In Eq. (84), fora photon the 4-momentum components in the t and r di-rections have to have the same signs, so if η is continuingforwards, then so must r. On the other hand, in the mas-sive particle case, Eq. (93), the t component is fixed insign (being m cosh Φ), while the r component (m sinh Φ)changes sign as Φ goes through zero. This is the originof the different behaviour as the FCB is crossed.

Our final task is to use this information to work outthe appropriate parameterisation in terms of proper timeafter the FCB. We see from the r′ result in Eq. (95),that since both sinh Φ and a change sign at the FCB,then r has to continue moving forward as a function ofproper time. Meanwhile the η derivative does changesign, meaning that after the FCB then proper time mustbe a decreasing function of conformal time. This ties inwith r in fact decreasing, since if proper time is decreas-ing then r will do so as well. We can regard this if wewish as the particle backtracking on its previous world-line, and now playing it out in reverse. Alternatively, itmay be that if we investigate the quantum mechanicalequations satisfied by the massive particle at the bound-ary, what we are seeing is that it has now become ananti-particle. This will be explored elsewhere. For themoment we content ourselves with giving the solutionsfor the particle position etc. in terms of τ once η hasbecome > 0. Writing X(τ) still as in Eq. (97), we find

Φ = ln (tanhX) , a = −P sinh(2X)

m

η =m

PH∞ sinh(2X), r = r1 +

m (1− coth(2X))

PH∞,

(101)for the background and geodesic parameters in terms ofτ . Here we have assumed Φ < 0 and that Φ goes throughthe value −Φ0 (with Φ0 still > 0) at τ = τ0, while rreaches r1 at the FCB as before. Note as just discussed,as the particle moves forward in conformal time afterthe FCB is reached, the proper time is dropping backdown from +∞, and retracing its values, and those ofr, before the FCB. We discuss some of these featuresagain, in relation to the different behaviour of matterand radiation perturbations, later.

3. GR translation

For those readers who would like to see the main resultsof the previous two subsections verified using standardGR, we show here how to recover the important aspects,at least up to issues of sign, using a metric rather than‘tetrad’ approach.

Let us consider wholly radial motion so that the por-tion of the FRW metric we need to consider is just

ds2 = a2(η)(dη2 − dr2), (102)

and for our model of the FCB we assume that

a = − 1

H∞η, (103)

as above, in Eq. (89).Since the metric in Eq. (102) is independent of space

r, the downstairs spatial component of the four-velocityur is conserved. The relativistic invariant also holds

uµuµ = a2(uη2 − ur2) = U2, (104)

20

where U = 1 for matter and U = 0 for photons. Theaffine parameter/proper time τ is implicitly defined by

uη =dη

dτ, ur =

dr

dτ. (105)

Combining the fact that ur = a2ur is a constant ofthe motion with Eqs. (103) to (105) gives the differentialequations (

dη

dτ

)2

−(dr

dτ

)2

= H2∞η

2U2 (106)

dr

dτ= urH

2∞η

2. (107)

For matter (U = 1), Eqs. (106) and (107) can be solvedin terms of our existing solutions from Sec. IV A 2 as

η = ± 1

urH∞ sinh(2X),

r − r1 −1

urH∞= −coth(2X)

urH∞

⇒ η2 =

(r − r1 −

1

urH∞

)2

− 1

u2rH

2, (108)

where we identify ur = P/m, X is as given in Eq. (97)and r1 is the radial coordinate of the particle when itreaches the FCB.

Eq. (108) can also be recovered by combiningEqs. (106) and (107) to give:(

dr

dη

)2

=η2

η2 + 1u2rH

2

. (109)

These results are consistent with the observations in theprevious subsections, showing that in the massive particlecase, r is a hyperbola in conformal time η.

For photons (U = 0), we take τ to be the same as theλ in Eq. (90). This means ur should be identified withthe photon energy E, and we find Eqs. (106) and (107)are solved by

η = ± 1

H2∞urτ

, and r − r1 = − 1

H2∞urτ

. (110)

We see that in all cases, η is only determined up to asign, as we might expect in this metric-based approach,which treats both directions in time equally.

B. Interpretation of the symmetry conditions

So far we have not been very clear about the natureof the universe beyond the future conformal boundary.In particular we have not been clear as to whether it issome new ‘aeon’, of the type discussed by Penrose, orperhaps instead a retracing of the evolution of our actualuniverse, but in a time-reversed direction. Some support

for the latter (less radical) viewpoint comes from the factthat it seems natural to take the relation

dη =1

adt, (111)

to be operating after we reach the FCB, in the sameway as before (though not inevitable, since the definingrelation in the metric is the squared one dt2 = a2dη2).In this case, since a flips sign after the FCB, and weare considering conformal time to be marching straightthrough the FCB, then the proper time of freely fallingobservers, t, will start going backwards from this point.In this view, therefore, conformal time provides a ‘doublecover’ of the universe’s evolution over the period betweenthe big bang and the FCB, rather than exploring somegenuinely new period beyond it. Equivalently, one couldconsider the properties of the future conformal boundaryto impose reflecting boundary conditions.

This is a very different philosophical picture, butadopting one or another picture does not change our con-clusions as regards the symmetry requirements we needto place on the perturbations. The important point isthat since the perturbations pass through the FCB withno apparent effects this means we have to make sure theyremain finite at all times, including in the ‘double cover’region of the conformal time/cosmic time evolution. Aswe have seen this then requires either symmetry or an-tisymmetry at the FCB itself, and hence leads to therequirements on K we have been discussing.

This picture potentially changes a bit when one consid-ers perturbations in a matter dominated universe, whichwe turn to next. This is because, as we will see, in somecircumstances it is possible to discuss a genuinely differ-ent time evolution history beyond the FCB than beforeit in this case. However, this hinges on questions aboutpositivity of matter energy density, and in the simplestversion of this, for which we work out the results in de-tail, the background time evolution after the FCB is thetime reverse of that before, and so a simple picture interms of a ‘double cover’, as just described, is sufficient.

V. SOME RESULTS FOR CDMPERTURBATIONS

We assume a simple pressureless fluid, and repeat theanalysis for this. Our first job is to find the analyticsolution for the background equations. We substitute

ρ =3Cs3

8πG, (112)

in the Einstein equations this time, and put C = Λ/3so that a = 1 at matter/vacuum energy density equal-ity (although this presumably does not happen halfwaythrough the conformal time development this time). Wealso have a time unit choice corresponding to Λ = 1.These choices yield the equations

3s2 = 1 + s3, and 3s2 − 2ss = 1. (113)

21

FIG. 22. Plot of Newtonian potential Φ as a function of con-formal time for normalised wave number K = 10 for CDMfluctuations.

Again one can verify that the derivative for the first equa-tion is compatible with the second. The difference of thetwo yields the simple equation

s = 12s

2. (114)

We can solve these equations via a Weierstrass ellipticfunction as follows:

s = 12℘

(η; 0,− 1

432

), (115)

where in the notation ℘(z; g1, g2), g1 and g2 are the in-variants being used, and we have chosen a possible offsetin η so that the double pole of the Weierstrass ellipticfunction occurs at the big bang.

The Φ equation in conformal time for this case is

Φ + 3aHΦ + Λa2Φ = 0. (116)

We note that unlike the equivalent equation in the ra-diation case, Eq. (3), there is no dependence on k, whichis due to the lack of pressure support for this type ofmatter. The analogue of Eq. (2) which gives V and δ interms of Φ, is

V =2k(aHΦ + Φ

)a2(3H2 − Λ)

,

δ = −2(

3H2a2Φ + 3ΦaH + k2Φ)

a2(3H2 − Λ).

(117)

In particular we see the δ equation is identical to theradiation case.

Some plots of these quantities in an example case aregiven in Figs. 22 to 24. The normalisation of theseis based on Φ starting with a value 1. (Of course thestarting Φ would in general be much smaller, and hencee.g. the density perturbation would have values much lessthan 1 in practice.)

FIG. 23. Plot of velocity perturbation V as a function ofconformal time for normalised wave number K = 10 for CDMfluctuations.

FIG. 24. Plot of density perturbation δ as a function of con-formal time for normalised wave number K = 10 for CDMfluctuations.

Now the essential thing we want to understand iswhat happens to these perturbations as the FCB is ap-proached, and in particular whether the perturbationscan ‘go through’ the FCB in a similar way to the radia-tion perturbations (which as we saw above, pass throughwithout noticing it). We note that in our current treat-ment, we are only working with one type of cosmic fluidat a time, and hence the question we are posing is notreally the full story — one would need to be consideringa universe with both radiation and CDM simultaneously,and with perturbations present in each, to get somethingwhich we can map onto our current universe. However,the simplified treatment here at least provides a start andmay show up problems that would still be present in afull treatment.

The first aspect to consider in relation to the behaviourat the FCB, is what happens to the evolution of the back-ground a itself. We have given an analytic solution abovefor s = 1/a, in Eq. (115). A plot of this function versus

22

FIG. 25. Plot of the inverse scale factor s = 1/a for a flat-Λuniverse with cold dark matter, showing the analytic solutiongiven in Eq. (115).

conformal time is given in Fig. 25. The plot may looksurprising, but we have plotted it for a full range of con-formal time η. The future conformal boundary is wheres first becomes 0, around η ≈ 4.8 (we will give an ex-act value of this shortly), and then s continues becomingmore negative until it flattens off around η ≈ 7.3, thenheads back towards 0, becoming positive again aroundη ≈ 9.7 and then heading towards +∞ in what looks likea ‘big crunch’. Perhaps the most surprising thing is thatthe solution is not symmetric about the point where itfirst becomes 0, but about a midpoint where there is a‘bounce’, at which |a| = 1/|s| reaches a (local) minimum.

We put modulus signs when describing s and a in thisregion, since of course they are negative. This raises aproblem with the interpretation of this solution that wasnot present for the radiation case. If we look at Eq. (112)for ρ, we see that since ρ is proportional to s3, then itswitches sign according to the sign of s, and if C stayspositive then ρ is negative when s is. On the other hand,in the radiation case, ρ is a fixed constant times s4, andthe latter does not change sign as s goes through 0, mean-ing that ρ stays positive after the FCB is reached.

The interpretation of a negative ρ is unclear. A pos-sible solution comes from scale-invariant gravity theory,where a so called ‘compensator field’ φ is introduced inthe Dirac Lagrangian to give the mass term the correctweight to be used in the theory. This was first introducedby Dirac himself [22] and is discussed further in [23] inthe context of eWGT (extended Weyl Gauge Theory).A solution within eWGT leading to a standard flat-Λcosmology is possible, in which the φ field goes to 0 atthe FCB and becomes negative afterwards. This meansthe effective ‘energy density’ due to matter remains non-negative despite s being negative. This idea is beyond theremit of the present paper, which uses only standard gen-eral relativity, so for the present we look at two possibleapproaches to the ρ problem which can be implemented

just in a GR context.First we can assume that something else, perhaps a

reinterpretation of what a negative scale factor itselfmeans, comes in to mean that a negative ρ is not aproblem for us, so that here we just extend analyticallythrough the FCB, making no attempt to prevent ρ < 0occurring.2

The second approach will be to take active steps toforce ρ to be positive after the FCB. Two obvious ways,which are equivalent in terms of their effect in the equa-tions, are to (i) switch the sign of the constant C inEq. (112) after the FCB or (ii), to change this equationinstead to

ρ =3C|s|3

8πG. (118)

while maintaining C positive throughout. One can con-sider this as the second branch of the conservation ofstress-energy equation for matter dρ/ρ = −3da/a, whichhas both Eqs. (112) and (118) as solutions.

Conceptually, we will implement this by seeking to joinas best we can the time development of the background,and other quantities, before the FCB using a positiveC, with time developments after the FCB correspondingto a negative C. We say ‘as best we can’ since as wewill shortly see it is not possible to carry this out whilstpreserving continuity of all derivatives, or indeed, in thecases of Φ and δ, any derivatives. We take continuityof the functions themselves as being the overriding aim,however, and will see that at least we can achieve this.

For the other approach mentioned above, where we donot seek to prevent ρ becoming negative, then the essenceof what we do will be to preserve analyticity throughout,for all quantities, and to rely on this to discover how theyin fact develop. It is in this spirit that Fig. 25 has beenplotted. This is for a single branch of the Weierstrasselliptic function and is completely analytic except at thetwo ends, which correspond to the big bang and the bigcrunch. We thus now seek to find analytic solutions forthe Newtonian potential Φ, and thence V and δ, whichcover this same region, and then after this we will look atthe other case, where C is taken to flip so as to preservethe density as being positive.

A. Analytic results for the CDM perturbations

We can seek to solve Eq. (116) for Φ in conformal timeby substituting for a = 1/s using s from Eq. (115) andfor H using the useful and simple relation

H = −dsdη. (119)

2 Within this approach we might take comfort from the fact thatwe eventually re-emerge into a universe with positive ρ and a,which seems easy to interpret, before heading for a big crunch,but we will see below that it is this latter period where in fact theproblems lie, and which a full eWGT treatment might remove.

23

FIG. 26. Plot of the Newtonian potential Φ for CDM fluc-tuations over the full range of conformal time η showing theanalytic solution given in Eq. (120).

This yields the following solution for Φ, where we haveadjusted constants so that Φ comes out of the big bangwith value 1, as previously:

Φ(η) = −1440

[2℘2

(η, 0,− 1

432

)+℘′

(η, 0,− 1

432

)ζ

(η, 0,− 1

432

)]℘

(η, 0,− 1

432

).

(120)Here ζ denotes the Weierstrass zeta elliptic function,whose derivative is −℘.

The resulting Φ plotted for the full range in conformaltime is shown in Fig. 26. We can see that Φ is not sym-metric about the mid-point at η ≈ 7.25 in this approach,and veers off towards being singular at the ‘big crunch’.

In fact there is good reason to believe (ANL, in prepa-ration), that the real limit in conformal time for this sys-tem is not at a big crunch, but before this at the secondcrossing of zero by s, at η ≈ 9.7. In the eWGT setupwith a compensator field φ mentioned above, this is thepoint where the combination of φ and ρ finally becomesnegative, meaning we would have genuine negative en-ergy density beyond this point. Furthermore, it turnsout to be the point of infinite redshift compared to theuniverse today. Hence for both these reasons, it is proba-bly the limit to which η should be taken, and we see thatΦ tends to 0 there, rather than becoming singular. Sincethere are no changes we can make to the wavenumberk to affect Φ, this is fortunate, and means (if we acceptthis interpretation) that despite Φ becoming singular ifwe did indeed go further to the right in η, in fact does notset any constraint on us as regards acceptable values of kor the general admissibility of CDM fluctuation modes, ifwe limit the physical range to before the point at whicha becomes positive again.

We now consider the equivalent plots for δ and V ,which are shown in Figs. 27 and 28. All of s, Φ andV go to 0 at the same point around η = 9.7, while δ ap-

FIG. 27. Plot of velocity potential V over the full range ofconformal time η, for CDM fluctuations with normalised wavenumber K = 10, as derived from equations (117) and (120).

FIG. 28. Same as for Fig. 27 but for density perturbations δ.

pears visually to have zero gradient near here, but moredetailed investigation shows that the point where locallyδ goes to zero only coincides with the other zeros in thehigh frequency limit, i.e. k → ∞. We can see why thisoccurs via looking at a general constraint on CDM fluc-tuations which we can derive by combining the derivativeof the expression for δ given in Eq. (117) with Eq. (116)for the second derivative of Φ. This yields

δ − 3Φ + kV = 0. (121)

Both δ and V get larger (in absolute terms) as k getslarger, with V including terms proportional to k and δterms proportional to k2, while Φ remains fixed. Thusin the high frequency limit we can ignore the Φ term inEq. (121), and therefore indeed find that δ and V go tozero at the same point.

24

B. Results for approach where density is keptpositive

We now consider results for the other approach men-tioned above, where a positive density is maintained be-yond the FCB via forcibly taking the modulus of ther.h.s. of Eq. (112) when s is negative, which is equivalentto flipping the sign of C there.

We will aim to do this whilst maintaining continuityin as many derivatives as possible in the functions we aredealing with. The first function to consider is s itself.With C flipped to −Λ/3, the equations it previously hadto satisfy, Eq. (113), are replaced by

3s2 = 1− s3, and 3s2 − 2ss,= 1 (122)

i.e. just the first order equation changes. Fairly naturally,minus the previous solution from Eq. (115) satisfies this,i.e. we could try

s = −12℘

(η; 0,− 1

432

). (123)

However, as one can see from Fig. 25, switching to thissolution at the first zero crossing of s will lead to a jumpin the first derivative of s. This is in fact not compatiblewith ρ being continuous as we can see by writing out theEinstein equations for the CDM case explicitly. Theseare:

3s2 − 8πGρ− Λ = 0,

3s2 − 2ss− Λ = 0.(124)

The first of these tells us that if ρ is continuous, thens must be as well, and the second then tells us that ifthis is so then s in fact has to be continuous as well, i.e.going down the list of derivatives the first that can bediscontinuous is the third one.

Examining Fig. 25, then suggests that since η does notappear explicitly in the equations, we could use the neg-ative of the s curve shown there for what happens afterthe FCB, but using the part starting after the second zerocrossing, not the first (the FCB corresponds to the firstcrossing of course). This will be compatible with the Ein-stein equations in terms of jump conditions provided thederivatives match at the FCB up to at least the secondderivative.

So we must now examine if this is the case. To dothis we need an explicit expression for where the FCBoccurs in η as well as expressions for the derivatives ofthe solutions Eqs. (115) and (123) at the appropriatepositions.

The relevant η where the first zero crossing occurs is

η0 =4π222/3

√3

9Γ3(

23

) ≈ 4.8573, (125)

and we can understand the derivatives of the two func-tions at the FCB by shifting to a new conformal timecoordinate

∆η = η − η0, (126)

FIG. 29. Plot of inverse scale factor s = 1/a(t) versus confor-mal time measured from the FCB, using the functions definedin Eq. (127).

and expressing the functions in terms of this. This leadsto the following explicit forms

s1 =

√3

216

(√3 + 36℘′

(∆η; 0,− 1

432

))℘2(∆η; 0,− 1

432

) ,

s2 = −√

3

216

(√3− 36℘′

(∆η; 0,− 1

432

))℘2(∆η; 0,− 1

432

) .

(127)

where s1 applies for ∆η < 0, i.e. before the FCB, and s2

applies for ∆η > 0, i.e. after the FCB.We can check what we need to about the derivatives

at the FCB by carrying out power series expansions at∆η = 0. This yields

s1 ≈ −1/3√

3∆η +1

72∆η4 − 1

9072

√3∆η7 +

1

435456∆η10,

s2 ≈ −1/3√

3∆η − 1

72∆η4 − 1

9072

√3∆η7 − 1

435456∆η10.

(128)We can see that our requirement is satisfied, since thefirst disagreement is at the ∆η4 term, meaning that thatfirst and second derivatives will match.

Our plot for s under these assumptions, is thus as givenin Fig. 29. It is interesting how the development after theFCB is very different from that shown in Fig. 25 (whichis for the case where we allow ρ to become negative),despite the fact the derivatives agree up to order 3 at theFCB.

Having decided how to continue s, we now consider Φ,V and δ. The important point will be their power seriesabout η = η0 and how well they match each side. Westart with Φ, since δ and V can continue to be derivedfrom this, using Eq. (117).

We could find the appropriate Φ for η < η0 if we couldshift the point of evaluation of each of the different typesof Weierstrass functions in Eq. (120) through an amountη0. (This works since the only dependence on η is through

25

these functions.) We can do this using the addition the-orems for Weierstrass functions as given e.g. on page 635of [24]. For example for the Weierstrass function ℘ wehave

℘(z1 + z2) =1

4

(℘′(z1)− ℘′(z2)

℘(z1)− ℘(z2)

)2

− ℘(z1)− ℘(z2).

(129)Using z1 = η and z2 = η0, and provided we know thevalues of ℘ and ℘′ at η0, then we can transfer the eval-uation point so that η becomes the ∆η = η − η0 definedabove. This is in fact how we achieved the expressionfor s1 given in Eq. (127). In the current case, for Φ, weneed to apply the equivalent process to ℘′

(η; 0,− 1

432

)and ζ

(η; 0,− 1

432

)as well. This requires knowing ζ at

η0 as well as ℘ and ℘′ there. Fortunately, since η0 cor-responds to a third of the total period of these ellipticfunctions, together with the fact that we can bring ourparticular problem into what is called ‘Equianharmonic’form (see [24] page 652) we are able to use the one-thirdperiod relations (p634 of [24]) and the results for ζ athalf periods in the equianharmonic case (p653 of [24]) todeduce the value of ζ at η0 which is

ζ

(η0; 0,− 1

432

)=

2π

3√

3

1

η0, (130)

with η0 being given as in Eq. (125).Inserting the shifted functions into Eq. (120) then gives

us the new Φ we require. This is quite complicated, sowe do not give the explicit expression here. However, theimportant aspect is the power series at η0, for which wefind

Φ(∆η < 0) = −5

6

3√

2 Γ(

23

)3π

∆η +5√

3

36∆η3

+25

432

3√

2 Γ(

23

)3√3

π∆η4 + . . .

(131)

We now need to consider Φ for ∆η > 0. For thiswe need to use Eq. (116) with the a(η) and H(η) cor-responding to the chosen s solution after η0, i.e. the s2 ofEq. (127). Furthermore, to give us the best set of oppor-tunities in carrying out matching for Φ (and the derivedδ and V ) we should use the most general possible solu-tion of Eq. (116) in this region. (We did not need to dothis for the Φ to the left of η0, since a particular solutionwith no free constants was picked out by the requirementthat Φ→ 1 as the big bang is approached.)

The needed general solution can be found as follows.First, we note the general solution for which Eq. (120)corresponds to a particular choice of constants, is

Φgen(η) = c1℘℘′ +

c29

(2℘2 + ℘′ζ

)℘, (132)

in an abbreviated notation where we have omitted the(η; 0,− 1

432

)on each function, and where c1 and c2 are

constants.

We then use the method just discussed to translatethis function through η0, so that it becomes a functionof ∆η. Finally, we need to make it work for the s2 formof s we are using beyond η0. By examining the form ofthe fundamental equation for Φ, Eq. (116), and by notingthat in Eq. (127) we can obtain s2 from s1, via

s2(∆η) = −s1(−∆η), (133)

we can show that the general Φ solution after η0 can befound from the general solution before η0 by flipping thesign of ∆η within each Weierstrass function. As withΦ(∆η < 0) the translation leads to quite a complicatedfunction, so we just give here the result for the powerseries at η0, which is

Φgen(∆η > 0) = − 1

15552

c23√

2 Γ(

23

)3+ 36c1π

π∆η

+c2√

3

93312∆η3 − 5

1119744

(c2

3√

2 Γ(

23

)3+ 36c1π

)√3

π∆η4 + . . .

(134)

Now comparing with Eq. (131) it may seem obviousthat we should match by taking c1 = 0 and c2 = 12960,since then Eq. (134) becomes

Φ(∆η > 0) = −5

6

3√

2 Γ(

23

)3π

∆η +5√

3

36∆η3

− 25

432

3√

2 Γ(

23

)3√3

π∆η4 +

11

864∆η6 + . . .

(135)and we have successfully matched up to third order.However, this ignores what happens with δ, which we willshortly show would be discontinuous with this choice ofconstants, and in fact what we need here is c1 = 0 andc2 = −12960, leading to

Φ(∆η > 0) =5

6

3√

2 Γ(

23

)3π

∆η − 5√

3

36∆η3

+25

432

3√

2 Γ(

23

)3√3

π∆η4 + . . .

(136)

This means that while Φ is continuous at η0, with value0, even the first derivatives do not match there.

So we now need to look at δ and V to understand thenecessity of the conclusion we have just reached. We canform these directly from Φ using Eq. (117), noting ofcourse that we need to use the appropriate a and H inthe different regimes. This leads to large expressions, butagain the important thing for us currently are the power

26

series either side of η0. We obtain

δ(∆η < 0) = −15π + 5 3√

2 (Γ (2/3))3k2√

3

3π

−5 3√

2 (Γ (2/3))3

2π∆η +

5 k2

6∆η2

+10 3√

2 (Γ (2/3))3k2 + 15π

√3

36π∆η3

+25 3√

2 (Γ (2/3))3√

3

144π∆η4 + . . .

(137)

and

δ(∆η > 0) =

(36π k2c1 + c2

3√

2 (Γ (2/3))3k2 + c2

√3π)√

3

7776π

−

(c2

3√

2 (Γ (2/3))3

+ 36 c1 π)

∆η

5184π− k2c2 ∆η2

15552

+

(2 c2

3√

2 (Γ (2/3))3k2 + 3 c2

√3π + 72π k2c1

)∆η3

93312π

−

(5 c2

3√

2 (Γ (2/3))3

+ 180 c1 π)√

3∆η4

373248π+ . . .

(138)while for V we get

V (∆η < 0) = −5/3 k∆η − 5/6k 3√

2 (Γ (2/3))3

∆η2

π

+5 k√

3∆η4

72+

5 k 3√

2 (Γ (2/3))3√

3∆η5

216π+ . . .

(139)and

V (∆η > 0) =kc2∆η

7776−k(

3√

2c2 (Γ (2/3))3

+ 36 c1π)

∆η2

15552π

+k√

3c2∆η4

186624−k(

3√

2c2 (Γ (2/3))3

+ 36 c1π)√

3∆η5

559872π+ . . .

(140)Now it is not obvious a priori that we have to require

δ to be continuous at η0, since the actual density per-turbation is δ × ρ, and of course ρ tends to 0 like s3

(which is ∝ ∆η3), at η0. However, we need to bring inEq. (121) which links the derivatives of δ and Φ withV . We have not shown it here, but in fact this rela-tion is the time component of the conservation relationfor the fluid stress-energy tensor, and therefore has to bestrictly obeyed. Suppose that δ was discontinuous at η0.We know from the above series that Φ and V are contin-uous there, since both have the value 0. Hence Eq. (121)would contain an unbalanced Dirac δ function from thederivative of (the density perturbation) δ. We thus knowδ must be continuous.

Since V is continuous, jumps in the derivatives of δand Φ must balance in this equation. This is automaticgiven our setup, however, since V is 0 at η0. In particularit can be verified that

δ|∆η=0+− δ|∆η=0−

= 3(

Φ|∆η=0+− Φ|∆η=0−

), (141)

FIG. 30. Plot of the Newtonian potential Φ for CDM fluctu-ations corresponding to the extension of the reciprocal scalefactor s through the FCB as shown in Fig. 29.

for all values of c1 and c2.

Where we can get an extra condition from, however,is from the following observation. As we have seen, theNewtonian potential is independent of wavenumber k,and we would certainly expect that this independenceshould persist after η0. In fact it is easy to show thatthe only values of c1 and c2 compatible with both thisrequirement and with δ being continuous, are c1 = 0 andc2 = −12960, as anticipated above. This also has thebenefit of making the first derivative of V match as well.

It may be worrying that this choice results in the firstderivative of Φ not matching, but as we have seen the im-portant thing is that jumps in this and the δ first deriva-tive must cancel, and this is automatic. Moreover, theactual metric perturbation, at linear order, is not Φ butsΦ, and the first derivative of this will match by virtueof being 0 on both sides.

We thus declare that we have found the unique contin-uation of all three quantities past η0, compatible withstress-energy tensor conservation and the requirementthat the matching for Φ is not dependent on k. Plotsof what these extended functions look like are shown inFigs. 30 to 32. We can see that what we have achievedhere is a time symmetric (or in the case of V time anti-symmetric), continuation of the three quantities, in whichthe development from the big bang to the FCB is reca-pitulated in reverse. In retrospect this is a fairly obvi-ous outcome, given that we have treated the reciprocalscale factor s in the same way, but it has been interest-ing to see how we have been forced to this conclusionby the logic of what needs to happen at the matchingpoint, rather than by desiring that the functions shouldreturn to non-singular values over their whole ranges af-ter the FCB. This means we can be confident that nonew boundary conditions are set by the FCB on CDMfluctuations — just the boundary conditions set at thebig bang, plus the requirements of SET conservation andlack of k-dependence of Φ after the FCB, are enough toguarantee the reverse time development found here.

27

FIG. 31. Plot of velocity potential V for K = 10 correspond-ing to the extension of the reciprocal scale factor s throughthe FCB as shown in Fig. 29.

FIG. 32. Plot of density perturbation δ for K = 10 cor-responding to the extension of the reciprocal scale factor sthrough the FCB as shown in Fig. 29. Note that despite itssymmetry, the derivative is non zero as η → 0, i.e. there is akink at η = 0 which is not visible on this scale.

Also, we have demonstrated that the method we haveused for extending s beyond the FCB in the case wherewe want to maintain the density as positive, is compati-ble with the boundary conditions set by the backgroundEinstein equations, which is a useful check. Of course,the fully analytic extension carried out in the first partsof Sec. V and based upon the extension for s shown inFig. 25, looks more mathematically natural, and has nodiscontinuities for any of the quantities, at any order of

derivative. However, in order for the matter density to bestill counted as corresponding to positive energy densitywe would probably need to introduce the compensatorscalar field φ discussed briefly above. We believe thisworks, but requires the underlying gravitational theoryto be extended beyond standard GR. This will be dis-cussed in future work.

VI. CONCLUSIONS

The ultimate fate of the universe suggested in this pa-per is of a profoundly different quality to conclusions thathave been proposed historically. Rather than a “BigFreeze” [25], “Big Rip” [26], “Big Crunch” [27], “BigBounce” [28] or conformal cycling [11], the equations ofour current concordance model show that our universepalindromes after infinite cosmic (but finite conformal)time. This new approach has, or can incorporate, el-ements of all of the above, and the double cover or re-flecting boundary condition interpretations have an anal-ogous quality to the Hartle-Hawking proposal for start ofthe universe [29], where there is no initial boundary.

The next step in researching these cosmologies is to gobeyond the analytic examples presented here, and to nu-merically synthesise the analysis presented in in Secs. IIIand V into a cosmology that more accurately correspondsto the universe we see today. This can then be comparedagainst modern observational data from the cosmic mi-crowave background [2] or supernovae [30]. An analysisin this regard can be found in our follow-up paper [14].Given such a profound modification to the underlyingcosmology, cosmologies such as these may have implica-tions for the tensions between measurements of cosmo-logical parameters at early and late times, such as thosediscussed in [31], and for which the Hubble constant isan example of great current interest.

ACKNOWLEDGMENTS

WJH is grateful to Gonville & Caius College for theirsupport via a Research Fellowship, and was also sup-ported by a Royal Society University Research Fellow-ship. DJB thanks the Cavendish Laboratory and TrinityCollege, Cambridge for their support during a Part IIIProject, is supported by STFC and Oriel College, Oxford,and acknowledges financial support from ERC Grant No693024.

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