Exact solution to perturbative conformal cosmology from recombination until the current era
Asanka Amarasinghe, Tianye Liu, Daniel A. Norman, Philip D. Mannheim
aa r X i v : . [ g r- q c ] J a n Exact solution to perturbative conformal cosmology from recombination until thecurrent era
Asanka Amarasinghe, Tianye Liu, Daniel A. Norman and Philip D. Mannheim
Department of Physics, University of Connecticut,Storrs, CT 06269, [email protected],[email protected],[email protected],[email protected] (Dated: January 5 2021)In a previous paper (P. D. Mannheim, Phys. Rev. D 102, 123535 (2020)) we studied cosmologicalperturbation theory in the cosmology associated with the fourth-order derivative conformal gravitytheory, and provided an exact solution to the theory in the recombination era. In this paper wepresent an exact solution that holds all the way from recombination until the current era.
I. INTRODUCTIONA. Motivation
A primary interest of cosmological research has been the study of cosmological fluctuations around a homogeneousand isotropic cosmic microwave background (see e.g. [1–5]). The focus in the main has been on the Einstein-gravity-based inflationary universe model [6], which leads to a concordance picture (see e.g. [7–9]) of a spatially flat universecomposed primarily of dark matter and dark energy. Given the lack to date of the detection of any dark mattercandidates, given the lack of understanding of the dark energy or cosmological constant 60 orders of magnitude finetuning problem, and given the presumption that the classical gravity treatment of the model that is made would notbe destroyed by uncontrollably infinite quantum gravitational radiative corrections [10], some candidate alternativeproposals have been advanced in the literature. In this paper we consider one specific alternative, namely conformalgravity (see [11–16] and references therein). As a candidate gravitational theory conformal gravity has been showncapable of eliminating the need for galactic dark matter not only by providing fits to a wide class of galactic rotationcurves without any need for dark matter but by doing so with universal galaxy-independent parameters [17–19], evenas, in contrast, one currently does have to introduce galaxy-dependent free parameters in dark matter fits. Also itwas shown that through its underlying local conformal symmetry (invariance under g µν ( x ) → e α ( x ) g µν ( x ) with aspacetime-dependent α ( x )) conformal gravity controls the cosmological constant without fine tuning. And with theconformal symmetry requiring that the gravitational sector coupling constant α g be dimensionless, the conformaltheory is renormalizable. With conformal gravity also being quantum-mechanically ghost free and unitary [20–23],conformal gravity provides a consistent quantum gravity theory in four spacetime dimensions.From the perspective of conformal gravity the dark matter, dark energy and quantum gravity problems are notthree separate problems at all. Rather, they all have a common cause, namely the extrapolation of Newton-Einsteingravity beyond its solar system origins. Consequently, they can all have a common solution, with conformal gravityendeavoring to provide such a solution through a different extrapolation of solar system wisdom. However, eventhough the conformal theory has successfully been applied to the homogeneous and isotropic cosmological backgroundby providing [14–16, 24] a horizon-free background cosmology with no flatness problem, while providing a very good,non-fine-tuned, dark-matter-free fit to the accelerating universe supernovae data of [25, 26], it still needs to be appliedto the fluctuations around that background. Such a study is now in progress, with the initial development of thecosmological perturbation theory that is required having been presented in general in [27–31]. In this paper we take afurther step by providing a new exact solution to the fourth-order derivative conformal gravity cosmological fluctuationequations that holds all the way from recombination until the current era. We had in fact already provided an exactsolution that holds at recombination itself [30], and in this paper we build on that study. In regard to conformalgravity we note also that various other studies of conformal gravity and of higher derivative gravity theories in generalcan be found in [32–46].In [30] we derived the conformal gravity cosmological fluctuation equations in all generality, for any backgroundcosmological geometry and any set of background matter sources. While these equations hold for any backgroundexpansion radius a ( t ) and any background spatial three-curvature k , in [30] we solved them exactly at recombination inthe case of most interest to conformal cosmology, namely negative k [14–16], a feature of the theory that we elaborateon below. To actually study the gravitational fluctuation equations we used the scalar, vector, tensor expansion ofthe fluctuation metric that was first introduced in [47] and [48] and then widely applied in perturbative cosmologicalstudies (see e.g. [49–54] and [1–5]). This expansion is based on quantities that transform as three-dimensionalscalars, vectors and tensors, and as such it is particularly well suited to Robertson-Walker geometries because suchgeometries have a spatial sector that is maximally three-symmetric. Even though the scalar, vector, tensor expansionis based on quantities whose transformation properties are defined with respect to three spatial dimensions rather thanfour spacetime dimensions, nonetheless using the scalar, vector, tensor basis leads to fluctuation equations that arecomposed of combinations of them that are fully four-dimensionally gauge invariant. They are thus very convenientfor cosmological fluctuation theory. Thus in this paper we shall follow the procedure laid out in [30].This paper is organized as follows. In Sec. I we describe the background gravitational and matter sectors. In Secs.II and III we consider fluctuations around the background using the scalar, vector, tensor fluctuation basis, and derivethe cosmological fluctuation equations of interest to us here. Some of the material presented in these sections hasalready been presented in [30], but is included here for completeness, and so that it can be adapted for the purposesof this paper. In Sec. IV we solve the fluctuation equations in the scalar sector in order to obtain the time behaviorof the scalar modes, doing so for both conformal time and comoving time. In Secs. V and VI we provide an analogousdiscussion for the vector modes and for the tensor modes. In Sec. VII we present the spatial behavior of the scalar,vector and tensor modes, and give normalization factors for the modes. Finally, in Sec. VIII we collect together allof our results, and give the full temporal and spatial behavior of the scalar, vector and tensor modes all the way fromrecombination until the current era. B. The Background Conformal Gravity Cosmology – Gravity Sector
Conformal gravity is a pure metric theory of gravity that possesses all of the general coordinate invariance andequivalence principle structure of standard gravity while augmenting it with an additional symmetry, local conformalinvariance, in which the action is left invariant under local conformal transformations on the metric of the form g µν ( x ) → e α ( x ) g µν ( x ) with arbitrary local phase α ( x ). Under such a symmetry a gravitational action that is to bea polynomial function of the Riemann tensor is uniquely prescribed, and with use of the Gauss-Bonnet theorem isgiven by (see e.g. [14]) I W = − α g Z d x ( − g ) / C λµνκ C λµνκ ≡ − α g Z d x ( − g ) / (cid:20) R µκ R µκ −
13 ( R αα ) (cid:21) . (1.1)Here α g is a dimensionless gravitational coupling constant, and C λµνκ = R λµνκ −
12 ( g λν R µκ − g λκ R µν − g µν R λκ + g µκ R λν ) + 16 R αα ( g λν g µκ − g λκ g µν ) (1.2)is the conformal Weyl tensor. (Here and throughout we follow the notation and conventions of [55].)With the Weyl action I W given in (1.1) being a fourth-order derivative function of the metric, functional variationwith respect to the metric g µν ( x ) generates fourth-order derivative gravitational equations of motion of the form [14] − − g ) / δI W δg µν = 4 α g W µν = 4 α g (cid:2) ∇ κ ∇ λ C µλνκ − R κλ C µλνκ (cid:3) = 4 α g (cid:20) W µν (2) − W µν (1) (cid:21) = T µν , (1.3)where the functions W µν (1) and W µν (2) (respectively associated with the ( R αα ) and R µκ R µκ terms in (1.1)) are given by W µν (1) = 2 g µν ∇ β ∇ β R αα − ∇ ν ∇ µ R αα − R αα R µν + 12 g µν ( R αα ) ,W µν (2) = 12 g µν ∇ β ∇ β R αα + ∇ β ∇ β R µν − ∇ β ∇ ν R µβ − ∇ β ∇ µ R νβ − R µβ R νβ + 12 g µν R αβ R αβ , (1.4)and where T µν is the conformal invariant energy-momentum tensor associated with a conformal matter source [56].Since W µν = W µν (2) − (1 / W µν (1) , known as the Bach tensor [57], is obtained from an action that is both generalcoordinate invariant and conformal invariant, in consequence, and without needing to impose any equation of motionor stationarity condition, W µν is automatically covariantly conserved and covariantly traceless and obeys ∇ ν W µν = 0, g µν W µν = 0 on every variational path used for the functional variation of I W . With the Weyl tensor vanishing ingeometries that are conformal to flat, in the conformal to flat Robertson-Walker and de Sitter geometries of interest tocosmology the background W µν is zero. Despite this, fluctuations around that background are not conformal to flat,with the fluctuating δW µν then being nonzero. However, with g µν W µν = 0 it follows that δg µν W µν + g µν δW µν = 0.Then with the cosmological background obeying W µν = 0, it follows that g µν δW µν = 0. Thus only nine of the tencomponents of the fluctuating δW µν are independent. C. The Background Cosmological Model
Since particles can only acquire mass in a conformal invariant theory by symmetry breaking, we introduce a scalarfield S ( x ) for this purpose. Even though it will not actually matter below since in the end the only thing we will needfrom the matter sector fields is a perfect fluid form for their energy-momentum tensor, for illustrative purposes wetake the matter sector fields to be represented by fermions, with the conformally invariant matter sector action thenbeing of the form I M = − Z d x ( − g ) / (cid:20) ∇ µ S ∇ µ S − S R µµ + λS + i ¯ ψγ c V µc ( x )[ ∂ µ + Γ µ ( x )] ψ − hS ¯ ψψ (cid:21) , (1.5)where h and λ are dimensionless coupling constants [58]. As such, the I M action is the most general curved spacematter action for the ψ ( x ) and S ( x ) fields that is invariant under both general coordinate transformations and the localconformal transformation S ( x ) → e − α ( x ) S ( x ), ψ ( x ) → e − α ( x ) / ψ ( x ), ¯ ψ ( x ) → e − α ( x ) / ¯ ψ ( x ), V aµ ( x ) → e α ( x ) V aµ ( x ), g µν ( x ) → e α ( x ) g µν ( x ). Variation of this action with respect to ψ ( x ) and S ( x ) yields the equations of motion iγ c V µc ( x )[ ∂ µ + Γ µ ( x )] ψ − hSψ = 0 , (1.6)and ∇ µ ∇ µ S + 16 SR µµ − λS + h ¯ ψψ = 0 . (1.7)We take the fermions (or whatever set of matter fields we may choose, even including the scalar field S ( x ) itself) toform a general background matter sector perfect fluid, and thus when the scalar field acquires a constant symmetrybreaking vacuum expectation value S the total background matter sector energy-momentum tensor is then of theform [14] T µν = 1 c [( ρ m + p m ) U µ U ν + p m g µν ] − S (cid:18) R µν − g µν R αα (cid:19) − g µν λS , (1.8)where the suffix m denotes matter. On taking the background geometry to be the comoving Robertson-Walker metric ds = c dt − a ( t ) (cid:20) dr − kr + r dθ + r sin θdφ (cid:21) = c dt − a ( t )˜ γ ij dx i dx j , (1.9)where ˜ γ ij is the background spatial sector metric, the background W µν then vanishes, so that the background T µν = 4 α g W µν then vanishes also. Since the background T µν does vanish, we can rewrite the T µν = 0 equation inthe instructive form 16 S (cid:18) R µν − g µν R αα (cid:19) = 1 c [( ρ m + p m ) U µ U ν + p m g µν ] − g µν λS . (1.10)We thus recognize the conformal cosmological evolution equation given in (1.10) as being of the form of none otherthan the cosmological evolution equation of the standard theory, viz. (on setting Λ = λS ) − c πG (cid:18) R µν − g µν R αα (cid:19) = 1 c [( ρ m + p m ) U µ U ν + p m g µν ] − g µν Λ , (1.11)save only for the fact that the standard G has been replaced by an effective, dynamically induced one given by G eff = − c πS , (1.12)viz. by an effective gravitational coupling that, as noted in [24], is expressly negative [59]. Conformal cosmologyis thus controlled by an effective gravitational coupling that is repulsive rather than attractive, and which becomessmaller the larger S might be. With G eff being negative, cosmological gravity is repulsive, and thus naturally leadsto cosmic acceleration.To see how central the negative sign of G eff is to cosmic acceleration we define¯Ω M ( t ) = 8 πG eff ρ m ( t )3 c H ( t ) , ¯Ω Λ ( t ) = 8 πG eff Λ3 cH ( t ) , ¯Ω k ( t ) = − kc ˙ a ( t ) , (1.13)where H = ˙ a/a . And on introducing the deceleration parameter q = − a ¨ a/ ˙ a , from (1.10) we obtain˙ a ( t ) + kc = ˙ a ( t ) (cid:0) ¯Ω M ( t ) + ¯Ω Λ ( t ) (cid:1) , ¯Ω M ( t ) + ¯Ω Λ ( t ) + ¯Ω k ( t ) = 1 ,q ( t ) = 12 (cid:18) p m ρ m (cid:19) ¯Ω M ( t ) − ¯Ω Λ ( t ) (1.14)as the background evolution equations of conformal cosmology.Without needing to specify any matter sector equation of state and without even needing to solve the theoryexplicitly at all, we are able to constrain q ( t ). Specifically, we note that since Λ represents the free energy that isreleased in the phase transition that generated S in the first place, Λ (and thus the scalar field coupling constant λ )is necessarily negative. Then with G eff also being negative the quantity ¯Ω Λ ( t ) is positive, i.e., the conformal theoryneeds a negative G eff in order to obtain a positive ¯Ω Λ ( t ). (In contrast, the standard model rationale for positiveΩ Λ = 8 πG Λ / c H is that since the Newtonian G is positive Λ has to be taken to be positive too.) Since ρ m and p m are associated with ordinary matter they are both positive. Thus ¯Ω M ( t ) is negative and ¯Ω Λ ( t ) is positive.Thus since G eff is negative it follows that q ( t ) is automatically negative, being so in every epoch. Consequently,conformal cosmology is automatically accelerating in every cosmological epoch without any adjustment or fine tuningof parameters ever being needed.While the standard model cannot accommodate a large Λ the conformal theory can since G eff can be much smallerthan G . In fact as S gets bigger Λ gets bigger too but G eff gets smaller, with ¯Ω Λ ( t ) self quenching. To see by how muchwe note that if we set Λ = − aT V /c ( V denotes vacuum) where T V is the large temperature at which the S generatingphase transition occurs, then with ¯Ω M ( t ) being of order aT /c , in the current era the ratio ¯Ω M ( t ) / ¯Ω Λ ( t )) = − T /T V is completely negligible. Moreover, since the temperature at recombination is only of order 1 eV , at recombination¯Ω M ( t ) / ¯Ω Λ ( t ) is negligible too. Thus we have to go to the very early universe in order to obtain a temperature atwhich ¯Ω M ( t ) / ¯Ω Λ ( t ) = − T /T V would not be negligible. In the very early universe we can use ρ m = 3 p m as theequation of state, and while massive matter would be non-relativistic at recombination, it would be irrelevant asto what equation of state we were to use for it since ¯Ω M ( t ) / ¯Ω Λ ( t ) is negligible at recombination. Since ¯Ω M ( t ) issuppressed from recombination onward, from recombination until the current era we can replace (1.14) by the simplerand more compact ˙ a ( t ) + kc = ˙ a ( t ) ¯Ω Λ ( t ) , ¯Ω Λ ( t ) + ¯Ω k ( t ) = 1 , q ( t ) = − ¯Ω Λ ( t ) . (1.15)In order to see by just how much ¯Ω Λ ( t ) might actually be self-quenched numerically in the conformal theory weneed to obtain a numerical bound for it. To this end we recall that study of galactic rotation curves in the conformaltheory enabled us to determine both the sign and magnitude of k , a quantity that is needed for (1.15), and thus forconformal cosmology. Specifically, solving the the fourth-order derivative W µν = 0 condition that would hold outsideof a static, spherically symmetric star (a situation in which the Weyl tensor is not zero), we find a potential of theform V ∗ ( r ) = − β ∗ c /r + γ ∗ c r/ /r potential we find a second potential,one that grows with distance. However, since all the other sources in the Universe would then themselves also beproducing potentials that grow with distance, they would thus also contribute to the net potential that would affectmaterial moving in a given galaxy. As shown in [12, 14, 17] this leads to two additional potentials, a linear potential γ c r/ − κc r coming from the inhomogeneitiesin that background. With the net potential V TOT ( r ) = − β ∗ c r + γ ∗ c r γ c r − κc r , (1.16)conformal gravity is able to fit the rotation curves of 138 galaxies [17–19], with the four parameters β ∗ , γ ∗ , γ and κ not varying from one galaxy to the next, while being fitted to the universal values β ∗ = 1 . × cm , γ ∗ = 5 . × − cm − ,γ = 3 . × − cm − , κ = 9 . × − cm − . (1.17)In contrast, we note that for the same 138 galaxy sample dark matter fits need 276 galaxy-dependent free parameters.For our purposes here we note that by transforming the comoving Hubble flow to the rest frame of any givengalaxy, the universal parameter γ is found [12] to be related to the spatial three-curvature of the Universe accordingto γ = ( − k ) / [60]. The three-curvature k thus has to be intrinsically negative, with the success of the rotationcurve fits showing that the motions of test particles in galaxies measure both the local and global gravitational fieldsand establish that in conformal cosmology the Universe must be topologically open. Since the net potential V TOT ( r )is generated by luminous matter alone, the view of conformal gravity is that the missing mass thought to be neededfor rotation curves is not in fact missing at all, it is the rest of the luminous matter in the Universe, and it has justbeen hiding in plain sight all along.On taking k to be negative we can now apply the conformal gravity theory to the accelerating universe data. With k being negative it follows that ¯Ω k ( t ) is positive. But ¯Ω Λ ( t ) is positive. Thus from (1.15) it follows that ¯Ω Λ ( t ) has tolie between zero and one. Hence no matter how big Λ itself might be, ¯Ω Λ ( t ), viz. the amount by which it gravitates,is self-quenched to not be anywhere near as big. Finally from (1.15) it also follows that with no fine tuning at all q ( t )is constrained to lie in the narrow negative range − ≤ q ( t ) ≤ G eff , Λ and k , we can now solve (1.15) for a ( t ) so as to determinethe evolution of the cosmological background in the region from recombination until the current time t , and in thisregion the requisite a ( t ) is given by [14] a ( t ) = ( − k ) / sinh( σ / ct ) σ / , (1.18)where σ = − λS = 8 πG eff Λ / c is positive. With such an a ( t ) we obtain¯Ω Λ ( t ) = tanh ( σ / ct ) , ¯Ω k ( t ) = sech ( σ / ct ) , q ( t ) = − tanh ( σ / ct ) , (1.19)and a luminosity distance redshift relation of the form [14] d L = − cH ( t ) (1 + z ) q " − (cid:18) q q (1 + z ) (cid:19) / , (1.20)where q = q ( t ) is the current era value of the deceleration parameter and H ( t ) is the current era value of theHubble parameter.Fitting the type 1A supernovae accelerating universe data with (1.20) gives a fit [14–16] that is comparable inquality with that of the standard model Ω M = 0 .
3, Ω Λ = 0 . q is fitted to the value − .
37, i.e., quite non-trivially found to be right in the allowed − ≤ q ≤ Λ ( t ) = 0 .
37, ¯Ω k ( t ) = 0 .
63. Since ¯Ω M is negligible no dark matter is needed, and since q and ¯Ω Λ = − q fall right in the allowed region, no fine tuning is needed either. The ability of the conformal gravity theory to fitthe accelerating universe data thus confirms that in conformal cosmology k is indeed negative. And since k = 0 isfavored in the standard concordance cosmological model, it is paramount to ascertain whether conformal cosmologicalfluctuations can support negative k , the ongoing objective of the current conformal cosmological studies [27–31].With tanh ( σ / ct ) being fitted to 0 .
37, we determine tanh( σ / ct ) = 0 .
61, sinh( σ / ct ) = 0 . σ / ct = 0 . H ( t ) = σ / c/ tanh( σ / ct ) = 1 . /t . With H = 72 km/sec/Mpc we obtain t = 4 . × sec, a perfectlyacceptable value for the age of the universe. Similarly, we obtain σ / c = 0 . × − sec − , σ / = 0 . × − cm − . Recalling that ( − k ) / = γ / . × − cm − , we obtain a ( t ) = 2 . × − , so the current era expansionradius itself is also small, something that will prove to be central to the study of this paper.If we extrapolate back to the recombination time t R we obtain a ( t R ) /a ( t ) = T /T R = O (10 − ). Consequently,with sinh( σ / ct ) = 0 .
77 we obtain sinh( σ / ct R ) = 0 . × − . Thus we can approximate sinh( σ / ct R ) by σ / ct R itself at recombination. Finally then, to one part in 10 for both ¯Ω Λ ( t R ) and ¯Ω k ( t R ) we have a ( t R ) = ( − k ) / ct R , ¯Ω Λ ( t R ) ≈ , ¯Ω k ( t R ) ≈ σ dropping out of a ( t R ), and with the numerical value of a ( t R ) being 2 . × − . As wesee, at recombination the conformal universe is curvature dominated. We thus recognize three epochs for conformalcosmology: radiation dominated early universe, curvature dominated recombination universe, cosmological constantdominated late universe. While there will always be a trace of ¯Ω M ( t ) in any non-early universe epoch, and whilenon-early universe propagating matter fields will respond to a geometry that they are not affecting in any substantialway, at recombination we see that a ( t R ) as given in (1.21) is independent not just of ¯Ω M ( t R ) but even of ¯Ω Λ ( t R ) aswell. It is the sheer simplicity of (1.21) that enabled us in [30] to obtain an exact solution to conformal cosmologyin the recombination era. In the following we will solve for the cosmology associated with the a ( t ) given in (1.18),and will again find an exact perturbative solution, one that because the current era a ( t ) is small, actually holdsperturbatively all the way from recombination until the current era.However, before doing so we should note that conformal models in which the scalar field is not an elementary fieldbut actually a vacuum expectation value h Ω | ¯ ψψ | Ω i of a fermion bilinear have also been considered [15, 16]. In thesemodels it is possible for the matter sources to make a more substantial contribution to cosmic expansion than inthe elementary scalar field case [61]. These dynamical models are not as straightforward to handle as the elementaryscalar field model and will be considered elsewhere. And indeed, it is the simplicity of a ( t ) = ( − k ) / sinh( σ / ct ) /σ / in the elementary scalar field model that enables us to solve the model completely analytically from recombinationonward, just as we now show. Moreover since a ( t ) is small in the entire region from recombination onward, we canuse perturbation theory to determine the gravitational fluctuations in that entire region. Now galaxies will eventuallygrow to large sizes. However while that will make δρ m /ρ m much larger than one, in the elementary scalar field modelthis overdensity will still be much smaller than the fluctuations in the gravitational field. Thus once we have solvedfor the gravitational fluctuations themselves (the objective of this paper), we can then include δρ m /ρ m type matterfield fluctuations as a small perturbation on them. Moreover, even in the event that matter fields were to contributesubstantially to conformal cosmological fluctuations (something our general formalism allows for even though it is notconsidered here), it is still of value (not just in conformal gravity but even in standard gravity) to ascertain what thepure gravitational contribution itself might be. II. THE FLUCTUATIONSA. Converting the Background to Conformal Time
While the above phenomenological discussion was developed for a specific background conformal cosmology with k <
0, we now discuss the fluctuation equations for arbitrary a ( t ), arbitrary k and arbitrary background mattersources. Rather than work in comoving time we have found it more convenient to work in conformal time. Thus ondefining τ = Z dta ( t ) , Ω( τ ) = a ( t ) , (2.1)we replace the background (1.9) by ds = Ω ( τ ) (cid:20) c dτ − dr − kr − r dθ − r sin θdφ (cid:21) = Ω ( τ )[ c dτ − ˜ γ ij dx i dx j ] , (2.2)with ˜ γ ij being the metric of the spatial sector, and with ( i, j, k ) = ( r, θ, φ ). In conformal time the background Einsteintensor is given by G = − k − c ˙Ω Ω − , G i = 0 , G ij = ˜ γ ij (cid:20) k − c ˙Ω Ω − + 2 c ¨ΩΩ − (cid:21) , R αα = − (cid:20) k + 1 c ¨ΩΩ − (cid:21) , (2.3)where the dot now denotes the derivative with respect to τ . In conformal time a generic background perfect matterfluid is described by T mµν = 1 c [( ρ m + p m ) U µ U ν + p m g µν ] , g µν U µ U ν = − , U = Ω − ( τ ) , U = − Ω( τ ) , U i = 0 , U i = 0 , (2.4)with covariant conservation condition ˙ ρ m + 3 ˙ΩΩ ( ρ m + p m ) = 0 . (2.5)For a conformal time radiation fluid with 3 p m = ρ m we obtain ρ m = A/ Ω , and for a non-relativistic fluid with p m = 0 we obtain ρ m = B/ Ω , viz. the same relations as obtained in comoving time. For conformal cosmology thebackground evolution equations are of the form4 α g W µν = 1 c [( ρ m + p m ) U µ U ν + p m g µν ] − S G µν − g µν λS = ∆ (0) µν , (2.6)with (2.6) serving to define the matter sector ∆ (0) µν . In a conformal to flat background geometry in which W µν = 0the background evolution equations take the form ∆ (0) µν = 0, viz.12 c S ( kc + ˙Ω Ω − ) + ρ m c Ω + Ω Λ = 0 , − c S ( kc − ˙Ω Ω − + 2 ¨ΩΩ − ) + p m c Ω − Ω Λ = 0 , c S ( kc + 2 ˙Ω Ω − − ¨ΩΩ − ) + Ω c ( ρ m + p m ) = 0 . (2.7)For ρ m = A/ Ω we obtain − S c Λ ˙Ω = " Ω + kS
4Λ + (cid:18) k S − A Λ c (cid:19) / Ω + kS − (cid:18) k S − A Λ c (cid:19) / . (2.8)While integrating (2.8) gives a somewhat intractable elliptic integral, in the non-early conformal gravity universe wecan ignore radiation and set A = 0, and with k and Λ both being negative then obtainΩ( τ ) = S ( k/ / sinh( − ( − k ) / cτ ) = − ( − k/σ ) / sinh(( − k ) / cτ ) . (2.9)To relate the conformal τ and the comoving t , from a ( t ) = ( − k/σ ) / sinh( σ / ct ) as given in (1.18) we set τ = Z dt ( − k/σ ) / sinh σ / ct = 1( − kc ) / log tanh( σ / ct/ , e ( − kc ) / τ = tanh( σ / ct/ , (2.10)as normalized so that τ = −∞ when t = 0 and τ = 0 when t = ∞ . (With the range of τ being negative, as given in(2.9) Ω( τ ) is positive everywhere within the range.) With Ω( τ ) = a ( t ), from (2.10) and a ( t ) = ( − k/σ ) / sinh( σ / ct )(2.9) then follows since σ = − /S . Finally, since at small comoving t the conformal time τ R goes to minus infinity,at recombination we can set Ω( τ R ) = 2 S ( k/ / exp[( − k ) / cτ R ] and a ( t R ) = ( − k ) / ct R . B. The Scalar, Vector, Tensor Basis for Fluctuations
In analyzing cosmological perturbations it is very convenient to use the scalar, vector, tensor (SVT) basis for thefluctuations as developed in [47] and [48]. In this basis the fluctuations are characterized according to how theytransform under three-dimensional rotations, and in this form the basis has been applied extensively in cosmologicalperturbation theory (see e.g. [49–54] and [1–5]. With the background metric being written with an overall conformalfactor Ω ( τ ) in (2.2), we shall take the fluctuation metric to also have an overall conformal factor, a particularlyconvenient choice in the conformal case since both the background W µν and the perturbative δW µν transform as W µν → Ω − W µν , δW µν → Ω − δW µν under a conformal transformation, so that, as shown in (2.20), the onlydependence of δW µν on Ω is in an overall Ω − conformal factor. We thus take the full metric to be of the form [62] ds = − ( g µν + h µν ) dx µ dx ν = Ω ( τ ) (cid:20) dτ − dr − kr − r dθ − r sin θdφ (cid:21) + Ω ( τ ) h φdτ −
2( ˜ ∇ i B + B i ) dτ dx i − [ − ψ ˜ γ ij + 2 ˜ ∇ i ˜ ∇ j E + ˜ ∇ i E j + ˜ ∇ j E i + 2 E ij ] dx i dx j i . (2.11)In (2.11) ˜ ∇ i = ∂/∂x i and ˜ ∇ i = ˜ γ ij ˜ ∇ j (with Latin indices) are defined with respect to the background three-spacemetric ˜ γ ij , and (1 , ,
3) = ( r, θ, φ ). And with˜ γ ij ˜ ∇ j V i = ˜ γ ij [ ∂ j V i − ˜Γ kij V k ] (2.12)for any three-vector V i in a three-space with three-space connection ˜Γ kij , the elements of (2.11) are required to obey˜ γ ij ˜ ∇ j B i = 0 , ˜ γ ij ˜ ∇ j E i = 0 , E ij = E ji , ˜ γ jk ˜ ∇ k E ij = 0 , ˜ γ ij E ij = 0 . (2.13)With the three-space sector of the background geometry being maximally three-symmetric, it is described by aRiemann tensor of the form ˜ R ijkℓ = k [˜ γ jk ˜ γ iℓ − ˜ γ ik ˜ γ jℓ ] . (2.14)As written, (2.11) contains ten elements, whose transformations are defined with respect to the background spatial sec-tor as four three-dimensional scalars ( φ , B , ψ , E ) each with one degree of freedom, two transverse three-dimensionalvectors ( B i , E i ) each with two independent degrees of freedom, and one symmetric three-dimensional transverse-traceless tensor ( E ij ) with two degrees of freedom. The great utility of this basis is that since the cosmologicalfluctuation equations are gauge invariant, only gauge-invariant scalar, vector, or tensor combinations of the compo-nents of the scalar, vector, tensor basis can appear in the fluctuation equations. In [31] it was shown that for thefluctuations associated with the metric given in (2.11) and with Ω( τ ) being an arbitrary function of τ and with k alsobeing arbitrary, the gauge-invariant metric combinations are α = φ + ψ + ˙ B − ¨ E, γ = − ˙Ω − Ω ψ + B − ˙ E, B i − ˙ E i , E ij , (2.15)for a total of six (one plus one plus two plus two) degrees of freedom, just as required since one can make fourcoordinate transformations on the initial ten fluctuation components. As we shall see below, the fluctuation equationswill explicitly depend on these specific combinations. Interestingly we note that even with nonzero k the gaugeinvariant metric combinations have no explicit dependence on k .Given the fluctuation basis we evaluate the fluctuation Einstein tensor, and obtain [29] δG = − kφ − kψ + 6 ˙ ψ ˙ΩΩ − + 2 ˙ΩΩ − ˜ ∇ a ˜ ∇ a B − − ˜ ∇ a ˜ ∇ a ˙ E − ∇ a ˜ ∇ a ψ,δG i = 3 k ˜ ∇ i B − ˙Ω Ω − ˜ ∇ i B + 2 .. ΩΩ − ˜ ∇ i B − k ˜ ∇ i ˙ E − ∇ i ˙ ψ − − ˜ ∇ i φ + 2 kB i − k ˙ E i − B i ˙Ω Ω − + 2 B i .. ΩΩ − + 12 ˜ ∇ a ˜ ∇ a B i −
12 ˜ ∇ a ˜ ∇ a ˙ E i ,δG ij = − .. ψ ˜ γ ij + 2 ˙Ω ˜ γ ij φ Ω − + 2 ˙Ω ˜ γ ij ψ Ω − − φ ˙Ω˜ γ ij Ω − − ψ ˙Ω˜ γ ij Ω − − .. Ω˜ γ ij φ Ω − − .. Ω˜ γ ij ψ Ω − − γ ij Ω − ˜ ∇ a ˜ ∇ a B − ˜ γ ij ˜ ∇ a ˜ ∇ a ˙ B + ˜ γ ij ˜ ∇ a ˜ ∇ a .. E + 2 ˙Ω˜ γ ij Ω − ˜ ∇ a ˜ ∇ a ˙ E − ˜ γ ij ˜ ∇ a ˜ ∇ a φ + ˜ γ ij ˜ ∇ a ˜ ∇ a ψ + 2 ˙ΩΩ − ˜ ∇ j ˜ ∇ i B + ˜ ∇ j ˜ ∇ i ˙ B − ˜ ∇ j ˜ ∇ i .. E − − ˜ ∇ j ˜ ∇ i ˙ E +2 k ˜ ∇ j ˜ ∇ i E − Ω − ˜ ∇ j ˜ ∇ i E + 4 .. ΩΩ − ˜ ∇ j ˜ ∇ i E + ˜ ∇ j ˜ ∇ i φ − ˜ ∇ j ˜ ∇ i ψ + ˙ΩΩ − ˜ ∇ i B j + 12 ˜ ∇ i ˙ B j −
12 ˜ ∇ i .. E j − ˙ΩΩ − ˜ ∇ i ˙ E j + k ˜ ∇ i E j − ˙Ω Ω − ˜ ∇ i E j + 2 .. ΩΩ − ˜ ∇ i E j + ˙ΩΩ − ˜ ∇ j B i + 12 ˜ ∇ j ˙ B i −
12 ˜ ∇ j .. E i − ˙ΩΩ − ˜ ∇ j ˙ E i + k ˜ ∇ j E i − ˙Ω Ω − ˜ ∇ j E i + 2 .. ΩΩ − ˜ ∇ j E i − .. E ij − E ij Ω − − E ij ˙ΩΩ − + 4 .. Ω E ij Ω − + ˜ ∇ a ˜ ∇ a E ij ,g µν δG µν = 6 ˙Ω φ Ω − + 6 ˙Ω ψ Ω − − φ ˙ΩΩ − −
18 ˙ ψ ˙ΩΩ − − .. Ω φ Ω − − .. Ω ψ Ω − − .. ψ Ω − + 6 kφ Ω − +6 kψ Ω − − − ˜ ∇ a ˜ ∇ a B − − ˜ ∇ a ˜ ∇ a ˙ B + 2Ω − ˜ ∇ a ˜ ∇ a .. E + 6 ˙ΩΩ − ˜ ∇ a ˜ ∇ a ˙ E − Ω − ˜ ∇ a ˜ ∇ a E + 4 .. ΩΩ − ˜ ∇ a ˜ ∇ a E + 2 k Ω − ˜ ∇ a ˜ ∇ a E − − ˜ ∇ a ˜ ∇ a φ + 4Ω − ˜ ∇ a ˜ ∇ a ψ. (2.16)For fluctuations in the matter field T mµν we obtain δT mµν = 1 c [( δρ m + δp m ) U µ U ν + δp m g µν + ( ρ m + p m )( δU µ U ν + U µ δU ν ) + p m h µν ] . (2.17)With g µν U µ U ν = − δg U U + 2 g U δU = 0 , (2.18)which entails that δU = −
12 ( g ) − ( − g g δg ) U = − Ω( τ ) φ, (2.19)with δU thus not being an independent degree of freedom. With δU i being a three-vector we shall decompose it intoits transverse and longitudinal parts as δU i = V i + ˜ ∇ i V , where now ˜ γ ij ˜ ∇ j V i = ˜ γ ij [ ∂ j V i − ˜Γ kij V k ] = 0. As constructed,in general we have 11 fluctuation variables, the six from the metric together with δρ m , δp m and the three δU i . Butwe only have ten fluctuation equations. Thus to solve the theory when there is both a δρ m and a δp m we will needsome constraint between δp m and δρ m . However, while this would be required if we want to obtain the generalsolution, as we had noted above, at recombination and onwards both δp m and δρ m are suppressed in the conformalcase, so no constraint between ρ m and p m is needed for our purposes here. Finally, we note that the fluctuation inthe cosmological constant term is just − λS h µν .The fluctuation δW µν in the Bach tensor W µν is of the form [28] δW = − ( ˜ ∇ a ˜ ∇ a + 3 k ) ˜ ∇ b ˜ ∇ b α,δW i = − ˜ ∇ i ( ˜ ∇ a ˜ ∇ a + 3 k ) ˙ α + 12Ω ( ˜ ∇ b ˜ ∇ b − ∂ τ − k )( ˜ ∇ c ˜ ∇ c + 2 k )( B i − ˙ E i ) , δW ij = − h ˜ γ ij ˜ ∇ a ˜ ∇ a ( ˜ ∇ b ˜ ∇ b + 2 k − ∂ τ ) α − ˜ ∇ i ˜ ∇ j ( ˜ ∇ a ˜ ∇ a − ∂ τ ) α i + 12Ω h ˜ ∇ i ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( ˙ B j − ¨ E j ) + ˜ ∇ j ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( ˙ B i − ¨ E i ) i + 1Ω h ( ˜ ∇ b ˜ ∇ b − ∂ τ − k ) + 4 k∂ τ i E ij . (2.20)We had noted above that since g µν δW µν does vanish, the tensor δW µν can only have nine independent components.With four coordinate invariances δW µν can only depend on five (rather than six) gauge invariant degrees of freedom,and as we see, they are α , B i − ˙ E i and E ij [63]. As we had also noted above, the only dependence of δW µν onΩ can be in an overall Ω − factor, and thus the five gauge invariant combinations on which it depends must haveno explicit dependence on Ω either (i.e., there must necessarily be five gauge invariant combinations that have noexplicit dependence on Ω). And in (2.15) we see that this is explicitly the case, and thus it has to be the Ω-dependent γ that does not appear in δW µν since only the Ω-independent ones can appear. Moreover, even though these samecombinations are gauge invariant even if we work in comoving coordinates [31], in comoving coordinates where there isno overall conformal factor in the comoving background metric given in (1.9), the B i − ∂ τ E i combination becomes the a ( t )-dependent B i − a ( t ) ∂ t E i . The lack of any explicit dependence of the gauge invariant combinations on k originatesin the fact that the Robertson-Walker metric itself can be written in a conformal to flat form (see e.g. [27]) in whichthe dependence in k can be put entirely in the conformal factor. Finally, we recall [31] that the gauge invariance ofthe combinations that appear in (2.15) can be established purely kinematically just by looking for combinations thatare left invariant under h µν → h µν − ∇ µ ǫ ν − ∇ ν ǫ µ . Consequently, these selfsame combinations must also appear inthe fluctuating matter sector energy-momentum tensor ∆ µν introduced below (just as they indeed do), and must evenbe the metric combinations that appear in perturbative Einstein gravity (just as they also in fact do, see e.g. [31]).From (2.6) we obtain background and fluctuation equations of the form4 α g W µν = 1 c [( ρ m + p m ) U µ U ν + p m g µν ] − S G µν − g µν Λ , α g δW µν = 1 c [( δρ m + δp m ) U µ U ν + δp m g µν + ( ρ m + p m )( δU µ U ν + U µ δU ν ) + p m h µν ] − S δG µν − h µν Λ . (2.21)It is convenient to define η = − α g S , R = − ρ m + c Λ) S , P = − p m − c Λ) S , δR = − δρ m S , δP = − δp m S . (2.22)The background and fluctuation equations then take the form ηW µν = G µν + 1 c [( R + P ) U µ U ν + P g µν ] = ∆ (0) µν , (2.23) ηδW µν = δG µν + 1 c [( δR + δP ) U µ U ν + δP g µν + ( R + P )( δU µ U ν + U µ δU ν ) + P h µν ] = ∆ µν , (2.24)with (2.23) and (2.24) serving to define ∆ (0) µν and ∆ µν . With the use of ∆ (0) µν = 0 (which follows here since W µν = 0),and with δG µν being given in (2.16), the components of ∆ µν have been determined in [29] and are of the form:∆ = 6 ˙Ω Ω − ( α − ˙ γ ) + δ ˆ R Ω + 2 ˙ΩΩ − ˜ ∇ a ˜ ∇ a γ, (2.25)∆ i = − − ˜ ∇ i ( α − ˙ γ ) + 2 k ˜ ∇ i γ + ( − Ω − + 2 .. ΩΩ − − k Ω − ) ˜ ∇ i ˆ V + k ( B i − ˙ E i ) + 12 ˜ ∇ a ˜ ∇ a ( B i − ˙ E i ) + ( − Ω − + 2 .. ΩΩ − − k Ω − ) V i , (2.26)∆ ij = ˜ γ ij (cid:2) Ω − ( α − ˙ γ ) − − ( ˙ α − ¨ γ ) − − ( α − ˙ γ ) + Ω δ ˆ P − ˜ ∇ a ˜ ∇ a ( α + 2 ˙ΩΩ − γ ) (cid:3) + ˜ ∇ i ˜ ∇ j ( α + 2 ˙ΩΩ − γ ) + ˙ΩΩ − ˜ ∇ i ( B j − ˙ E j ) + 12 ˜ ∇ i ( ˙ B j − ¨ E j ) + ˙ΩΩ − ˜ ∇ j ( B i − ˙ E i ) + 12 ˜ ∇ j ( ˙ B i − ¨ E i ) − .. E ij − kE ij − E ij ˙ΩΩ − + ˜ ∇ a ˜ ∇ a E ij , (2.27)0˜ γ ij ∆ ij = 6 ˙Ω Ω − ( α − ˙ γ ) − − ( ˙ α − ¨ γ ) −
12 ¨ΩΩ − ( α − ˙ γ ) + 3Ω δ ˆ P − ∇ a ˜ ∇ a ( α + 2 ˙ΩΩ − γ ) , (2.28) g µν ∆ µν = 3 δ ˆ P − δ ˆ R − .. ΩΩ − ( α − ˙ γ ) − − ( ˙ α − ¨ γ ) − − ˜ ∇ a ˜ ∇ a ( α + 3 ˙ΩΩ − γ ) , (2.29)where Ω R = 3 k + 3 ˙Ω Ω − , Ω P = − k + ˙Ω Ω − − − , ˙ R + 3 ˙Ω( R + P )Ω − = 0 ,α = φ + ψ + ˙ B − ¨ E, γ = − ˙Ω − Ω ψ + B − ˙ E, ˆ V = V − Ω ˙Ω − ψ,δ ˆ R = δR −
12 ˙Ω ψ Ω − + 6 .. Ω ψ Ω − − kψ Ω − = δR + ˙Ω − ˙ Rψ Ω = δR − R + P ) ψ,δ ˆ P = δP − ψ Ω − + 8 .. Ω ψ Ω − + 2 kψ Ω − − ... Ω ˙Ω − ψ Ω − = δP + ˙Ω − ˙ P ψ Ω . (2.30)(The first three expressions in (2.30) hold for the background and follow from ∆ (0) µν = 0.) With ηδW µν − ∆ µν beinggauge invariant and with δW µν being gauge invariant on its own, it follows that ∆ µν is gauge invariant too, and thusits dependence on the metric sector fluctuations must be solely on the metric combinations α , γ , B i − ˙ E i and E ij ,just as we see. Then since the metric sector α , γ , B i − ˙ E i and E ij are gauge invariant, from the gauge invariance of∆ µν , and as shown directly in [31], it follows that δ ˆ R , δ ˆ P , ˆ V and V i are gauge invariant too. We thus have expressedthe fluctuation equations entirely in terms of gauge invariant combinations without needing to specify any particulargauge. Given (2.20) and (2.25) - (2.27) the full conformal cosmological fluctuation equations take the form ηδW = − η ( ˜ ∇ a ˜ ∇ a + 3 k ) ˜ ∇ b ˜ ∇ b α = ∆ = 6 ˙Ω Ω − ( α − ˙ γ ) + δ ˆ R Ω + 2 ˙ΩΩ − ˜ ∇ a ˜ ∇ a γ, (2.31) ηδW i = − η ˜ ∇ i ( ˜ ∇ a ˜ ∇ a + 3 k ) ˙ α + η ( ˜ ∇ b ˜ ∇ b − ∂ τ − k )( ˜ ∇ c ˜ ∇ c + 2 k )( B i − ˙ E i )= ∆ i = − − ˜ ∇ i ( α − ˙ γ ) + 2 k ˜ ∇ i γ + ( − Ω − + 2 .. ΩΩ − − k Ω − ) ˜ ∇ i ˆ V + k ( B i − ˙ E i ) + 12 ˜ ∇ a ˜ ∇ a ( B i − ˙ E i ) + ( − Ω − + 2 .. ΩΩ − − k Ω − ) V i , (2.32) ηδW ij = − η h ˜ γ ij ˜ ∇ a ˜ ∇ a ( ˜ ∇ b ˜ ∇ b + 2 k − ∂ τ ) α − ˜ ∇ i ˜ ∇ j ( ˜ ∇ a ˜ ∇ a − ∂ τ ) α i + η h ˜ ∇ i ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( ˙ B j − ¨ E j ) + ˜ ∇ j ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( ˙ B i − ¨ E i ) i + η Ω h ( ˜ ∇ b ˜ ∇ b − ∂ τ − k ) + 4 k∂ τ i E ij = ∆ ij = ˜ γ ij (cid:2) Ω − ( α − ˙ γ ) − − ( ˙ α − ¨ γ ) − − ( α − ˙ γ ) + Ω δ ˆ P − ˜ ∇ a ˜ ∇ a ( α + 2 ˙ΩΩ − γ ) (cid:3) + ˜ ∇ i ˜ ∇ j ( α + 2 ˙ΩΩ − γ ) + ˙ΩΩ − ˜ ∇ i ( B j − ˙ E j ) + 12 ˜ ∇ i ( ˙ B j − ¨ E j ) + ˙ΩΩ − ˜ ∇ j ( B i − ˙ E i ) + 12 ˜ ∇ j ( ˙ B i − ¨ E i ) − .. E ij − kE ij − E ij ˙ΩΩ − + ˜ ∇ a ˜ ∇ a E ij . (2.33)In the conformal gravity theory these cosmological fluctuation equations are completely general, and hold for anypossible matter source and any possible a ( t ) and k .We had noted above that the only difference between the conformal gravity (1.10) and the standard Einstein gravity(1.11) was in the replacement of the Newtonian G by the conformal gravity G eff given in (1.12). We can thus treat both∆ µν (0) and ∆ µν as being generic to both theories. Consequently, Einstein gravity fluctuation theory can be recognizedas the η = 0 limit of the conformal gravity ηW µν − ∆ µν (0) = 0, ηδW µν − ∆ µν = 0 in which ∆ µν (0) = 0 and ∆ µν = 0.We should also add that the parameter α g is actually known to be negative [22, 64]. The parameter η = − α g /S is thus positive. This will prove to be a key feature of the development below as it will lead us to solutions to thefluctuation equations that oscillate in time rather than grow or decay exponentially. III. THE CONFORMAL GRAVITY DECOMPOSITION THEOREM
As constructed, the ten ηδW µν = ∆ µν fluctuation equations mix the perturbative fluctuation quantities. In Einsteingravity a similar situation arises and there one appeals to the decomposition theorem to break the fluctuation equations1up into separate scalar, vector and tensor sector equations. By imposing boundary conditions at both r = ∞ and r = 0 a proof of this theorem was given in [29] for Einstein gravity. And using the same boundary conditions it wasshown in [30] that the decomposition theorem also holds for the conformal gravity fluctuations of interest to us inthis paper. Thus for the conformal gravity case the relation ηδW µν = ∆ µν breaks up into the ten relations [30] − η ( ˜ ∇ a ˜ ∇ a + 3 k ) ˜ ∇ b ˜ ∇ b α = 6 ˙Ω Ω − ( α − ˙ γ ) + δ ˆ R Ω + 2 ˙ΩΩ − ˜ ∇ a ˜ ∇ a γ, (3.1)12 ( ˜ ∇ c ˜ ∇ c + 2 k ) η Ω ( ˜ ∇ b ˜ ∇ b − ∂ τ − k )( B i − ˙ E i )= 12 ( ˜ ∇ c ˜ ∇ c + 2 k )( B i − ˙ E i ) + ( − Ω − + 2 .. ΩΩ − − k Ω − ) V i , (3.2) η Ω h ( ˜ ∇ b ˜ ∇ b − ∂ τ − k ) + 4 k∂ τ i E ij = − .. E ij − kE ij − E ij ˙ΩΩ − + ˜ ∇ a ˜ ∇ a E ij . (3.3) − η ˜ ∇ a ˜ ∇ a ( ˜ ∇ b ˜ ∇ b + 2 k − ∂ τ ) α = 2 ˙Ω Ω − ( α − ˙ γ ) − − ( ˙ α − ¨ γ ) − − ( α − ˙ γ ) + Ω δ ˆ P − ˜ ∇ a ˜ ∇ a ( α + 2 ˙ΩΩ − γ ) , (3.4) η ( ˜ ∇ a ˜ ∇ a − ∂ τ ) α = α + 2 ˙ΩΩ − γ, (3.5) − η ( ˜ ∇ a ˜ ∇ a + 3 k ) ˙ α = − − ( α − ˙ γ ) + 2 kγ + ( − Ω − + 2 .. ΩΩ − − k Ω − ) ˆ V , (3.6) η ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( ˙ B i − ¨ E i ) = ˙ΩΩ − ( B i − ˙ E i ) + 12 ( ˙ B i − ¨ E i ) , (3.7)viz. four one-component scalar equations [(3.1), (3.4), (3.5), (3.6)], two two-component vector equations [(3.2), (3.7)],and one two-component tensor equation [(3.3)]. We note that all of these equations are both gauge invariant andexact without approximation, and hold in any cosmological epoch.As written, these ten equations involve 11 degrees of freedom, the six associated with the metric, viz. α , γ , B i − ˙ E i and E ij , and the five associated with the matter fluctuations, viz. δ ˆ R , δ ˆ P , ˆ V and V i . We note that the tensorequation (3.3) does not involve the fluctuating matter source at all. Thus once we specify a form for Ω( τ ), somethingthat would involve the background matter source but not the fluctuating one, we can then solve the tensor sectorcompletely and will do so below.Additionally, while the two vector sector equations do involve the matter fluctuation V i , its behavior is highlyconstrained. Specifically we can write (3.7) as ∂∂τ (cid:20) η ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( B i − ˙ E i ) − Ω ( B i − ˙ E i ) (cid:21) = 0 , (3.8)and thus integrate it to η ( ˜ ∇ a ˜ ∇ a − k − ∂ τ )( B i − ˙ E i ) − Ω ( B i − ˙ E i ) = C i , (3.9)where the integration constant C i depends solely on the spatial coordinates. Then, if we apply ( ˜ ∇ c ˜ ∇ c + 2 k ) to (3.9)and compare with (3.2) we obtain a relation that involves V i alone, viz.( ˜ ∇ c ˜ ∇ c + 2 k ) C i = 2Ω ( − Ω − + 2 .. ΩΩ − − k Ω − ) V i . (3.10)Then, since C i is independent of τ , the τ dependence of V i is completely fixed, to be of the unique form Ω − ( − Ω − +2 .. ΩΩ − − k Ω − ) − . Consequently, V i is not a dynamical variable, and we shall thus set it to zero in the following.With V i vanishing, C i then obeys ( ˜ ∇ c ˜ ∇ c + 2 k ) C i = 0. Now in [29, 30] we showed that the equation ( ˜ ∇ c ˜ ∇ c + 2 k ) C i = 02with any vector C i had no non-trivial solutions at all that were well behaved at both r = ∞ and r = 0. We thusconclude that C i is zero, with (3.9) then reducing to η Ω ( ˜ ∇ b ˜ ∇ b − ∂ τ − k )( B i − ˙ E i ) − ( B i − ˙ E i ) = 0 . (3.11)Alternatively, we could set C i = 0 as an initial condition for the integration of (3.8), and then V i = 0 would followfrom (3.10) unless − Ω − + 2 .. ΩΩ − − k Ω − is zero. And while we will actually take − Ω − + 2 .. ΩΩ − − k Ω − to be zero below (with (2.9) then being its solution), in such a case (3.10) would still lead to to C i = 0. Thus eitherway we finish up with (3.11). Thus in the following we only need to solve (3.11), while noting that like the tensorsector (3.3), (3.11) also has no dependence on the matter fluctuations.Thus the only fluctuation equations that do involve the matter fluctuations are all in the scalar sector. At thispoint one cannot in general proceed without further information since while there are four scalar sector equationsthere are five scalar sector degrees of freedom, α , γ , δ ˆ R , δ ˆ P and ˆ V . To address this issue one looks for some relationbetween δp m and δρ m . If as is standard we set δp m /δρ m = v and take as equation of state p m = wρ m where w isconstant, then from their definitions it follows that δ ˆ P − wδ ˆ R = − S ( δp m − wδρ m ) = − S ( v − w ) δρ m . (3.12)Once we specify such w and v , we can then in principle solve the scalar sector completely in any backgroundcosmology in any cosmological epoch [65]. However, as we had noted above, the specific phenomenological structureof the conformal cosmology of interest to us in this paper is such that the matter sector fluctuation are negligiblefrom recombination onwards, with, as per (2.7), the expansion radius Ω( τ ) then obeying − Ω − + ¨ΩΩ − − k = 0.Thus with this being case the full set of fluctuation equations is then of the form − η ( ˜ ∇ a ˜ ∇ a + 3 k ) ˜ ∇ b ˜ ∇ b α − Ω − ( α − ˙ γ ) − − ˜ ∇ a ˜ ∇ a γ = 0 , (3.13) − η ˜ ∇ a ˜ ∇ a ( ˜ ∇ b ˜ ∇ b + 2 k − ∂ τ ) α − Ω − ( α − ˙ γ ) + 2 ˙ΩΩ − ( ˙ α − ¨ γ )+ 4 ¨ΩΩ − ( α − ˙ γ ) + ˜ ∇ a ˜ ∇ a ( α + 2 ˙ΩΩ − γ ) = 0 , (3.14) η ( ˜ ∇ a ˜ ∇ a − ∂ τ ) α − α − − γ = 0 , (3.15) − η ( ˜ ∇ a ˜ ∇ a + 3 k ) ˙ α + 2 ˙ΩΩ − ( α − ˙ γ ) − kγ = 0 , (3.16) η Ω ( ˜ ∇ b ˜ ∇ b − ∂ τ − k )( B i − ˙ E i ) − ( B i − ˙ E i ) = 0 , (3.17) η Ω h ( ˜ ∇ b ˜ ∇ b − ∂ τ − k ) + 4 k∂ τ i E ij + ¨ E ij + 2 kE ij + 2 ˙ E ij ˙ΩΩ − − ˜ ∇ a ˜ ∇ a E ij = 0 , (3.18)and we note that because − Ω − + 2 .. ΩΩ − − k Ω − would now be zero, the ˆ V term in (3.6) has dropped outidentically in (3.16). These then are the equations that we need to solve. IV. SCALAR SECTOR SOLUTION FROM RECOMBINATION UNTIL THE CURRENT ERA
For the scalar sector we only have two independent degrees of freedom at recombination, α and γ , but we havefour equations, (3.13), (3.14), (3.15) and (3.16). There must thus be two relations between them, and on setting − Ω − + ¨ΩΩ − − k = 0, we find that the left-hand sides of these equations obey ddτ (cid:16) − (3 .
16) + (3 . (cid:17) = (cid:16) ˜ ∇ b ˜ ∇ b + 3 k − Ω − (cid:17) (3 . − ˙ΩΩ − (3 .
13) + 2 ˙ΩΩ − ( ˜ ∇ b ˜ ∇ b + 3 k )(3 . , (cid:18) ddτ + 2 ˙ΩΩ − (cid:19) (3 .
16) = ( ˜ ∇ b ˜ ∇ b + 2 k )(3 .
15) + (3 .
14) (4.1)identically. And with the right-hand sides of (3.13), (3.14), (3.15) and (3.16) vanishing, for the two remaining relationswe note first that3 ˙ΩΩ − (3 .
16) + (3 . − ( ˜ ∇ b ˜ ∇ b + 3 k )(3 .
15) = ( ˜ ∇ b ˜ ∇ b + 3 k ) h η Ω (¨ α − − ˙ α − ˜ ∇ b ˜ ∇ b α ) + α i = 0 , (4.2)and can thus set [66] η [¨ α − − ˙ α − ˜ ∇ b ˜ ∇ b α ] = − Ω α, (4.3)to thereby fix α . And secondly, from (3.15) we can determine γ according to γ = Ω2 ˙Ω h η ( ˜ ∇ a ˜ ∇ a − ∂ τ ) α − α i . (4.4)As constructed, α will depend on both the spatial coordinates and τ . As discussed in [29, 30] and in more detailbelow, the spatial sector separates according to [ ˜ ∇ b ˜ ∇ b + ( − k ) A S ] α = 0, where ( − k ) A S is a separation constant. Onintroducing ρ = ( − k ) / τ , and setting κ = ση , then with Ω( τ ) = − ( − k/σ ) / / sinh(( − k ) / τ ) as per (2.9), followingthe separation of variables the time dependence of α , viz. α ( ρ ), is fixed by (cid:18) d dρ + 2 cosh ρ sinh ρ ddρ + A S + 1 κ sinh ρ (cid:19) α ( ρ ) = 0 , (4.5)where κ is positive since both σ and η are positive [67].To solve (4.5) we first note that when η and A S are real (below we will see that A S is real and greater than one),(4.5) is a real second-order derivative equation. Thus its solutions are either real or in complex conjugate pairs.Consequently we can always find real solutions, either ones that are already real or any complex solution plus itscomplex conjugate. And it is understood that whenever we obtain a complex solution it is always to be accompaniedby its complex conjugate. To find the solutions we set z = cosh ρ so that ddρ = ( z − / ddz , d dρ = ( z − d dz + z ddz . (4.6)Then (4.5) becomes (cid:20) ( z − d dz + 3 z ddz + A S + 1 /κz − (cid:21) α = 0 . (4.7)Next we set α = ( z − − / β and obtain (cid:20) ( z − d dz + 2 z ddz + A S −
34 + 1 /κ − / z − (cid:21) β = 0 . (4.8)This is now in the standard form of the associated Legendre function equation (cid:20) ( z − d dz + 2 z ddz − ζ ( ζ + 1) − µ z − (cid:21) β = 0 , (4.9)where, on setting ν = A S − ζ = − ± (1 − A S ) / = − ± iν, µ = 14 − κ , (4.10)with associated Legendre solutions P µζ ( z ), Q µζ ( z ) with z = cosh ρ > P µζ ( z ) = 1Γ(1 − µ ) (cid:18) z + 1 z − (cid:19) µ/ F ( − ζ, ζ + 1; 1 − µ ; (1 − z ) / , (4.11) Q µζ ( z ) = e iµπ Γ( ζ + µ + 1)Γ(1 / ζ +1 Γ( ζ + 3 /
2) ( z − µ/ z − ζ − µ − F (( ζ + µ + 2) / , ( ζ + µ + 1) / ζ + 3 /
2; 1 /z ) . (4.12)4 A. Conformal Time Scalar Sector Solutions
Thus in conformal time the first solution for α ( ρ ) is of the form α ( ρ ) = 1sinh / ρ P µζ (cosh ρ ) = 1sinh / ρ − µ ) coth µ ( ρ/ F ( − ζ, ζ + 1; 1 − µ ; − sinh ( ρ/ . (4.13)We thus obtain the conformal time α ( ρ ) in a completely closed form. Once we have α we can get γ from (3.15), andwith, as per (2.9), Ω = − ( − k/σ ) / / sinh ρ we have − κ ρ ( ν + 1 + 3 ∂ ρ ) α ( ρ ) − α ( ρ ) = − ρ sinh ρ ( − k ) / γ ( ρ ) . (4.14) B. Comoving Time Scalar Sector Solutions
It is also useful to write everything in comoving time. In comoving time t (4.3) becomes a d αdt − a dadt dαdt + ( − k ) A S α + a αη = 0 . (4.15)Setting α ( t ) = σ / ( − k ) − / a / δ ( t ) we find that δ obeys a d δdt + 2 a dadt dδdt + ( − k ) A S δ + a δη + 32 a d adt δ − (cid:18) dadt (cid:19) δ = 0 . (4.16)On setting ξ = σ / t and, as per (1.18), a = ( − k/σ ) / sinh ξ , so that α = sinh / ξδ , we obtain¨ δ + 2 cosh ξ sinh ξ ˙ δ + A S − / ξ δ + (cid:20) κ + 34 (cid:21) δ = 0 , (4.17)where the dot now denotes d/dξ and as before κ = ση . We now proceed as in (4.5) only with the replacements A S → κ + 34 , κ → A S − , (4.18)so that now, with A S − ν we obtain ζ = − ± (cid:18) − κ (cid:19) / , µ = 1 − A S = − ν . (4.19)Thus we have α ( ξ ) = sinh ξP µζ (cosh ξ ) = sinh ξ − µ ) coth µ ( ξ/ F ( − ζ, ζ + 1; 1 − µ ; − sinh ( ξ/ . (4.20)We thus obtain the comoving time α ( ξ ) in a completely closed form.For γ , (4.4) is given in comoving time by2 dadt γ = η a (cid:18) ˜ ∇ a ˜ ∇ a − a ∂ ∂t − a dadt ∂∂t (cid:19) α − α, (4.21)and thus in terms of ξ = σ / t and a = ( − k/σ ) / sinh ξ we find that the comoving time dependence of γ is given by − κ ξ (cid:18) A S + 3 sinh ξ ∂ ∂ξ + 3 sinh ξ cosh ξ ∂∂ξ (cid:19) α ( ξ ) − α ( ξ ) = 2 γ ( ξ )( − k ) / cosh ξ, (4.22)5where the comoving time α ( ξ ) is given in (4.20). To evaluate γ ( ξ ) we need to evaluate the first and second timederivatives of P µζ ( z ), where z = cosh ξ . To this end we change the variable from ξ to z according to ∂ ξ = ( z − / ∂ z , ∂ ξ = ( z − ∂ z + z∂ z . (4.23)Thus with sinh ξ = ( z − / we obtainsinh ξ∂ ξ + sinh ξ cosh ξ∂ ξ = ( z − ∂ z + 2 z ( z − ∂ z . (4.24)Thus from (4.9) we obtain (cid:2) sinh ξ∂ ξ + sinh ξ cosh ξ∂ ξ (cid:3) P µζ ( z ) = sinh ξ (cid:20) ζ ( ζ + 1) + µ sinh ξ (cid:21) P µζ ( z ) . (4.25)Then with P µζ ( z ) obeying the functional relations ddz P µζ ( z ) = ( z − − h ( ζ − µ + 1) P µζ +1 ( z ) − ( ζ + 1) zP µζ ( z ) i , ξ ddξ P µζ (cosh ξ ) = 1sinh ξ h ( ζ − µ + 1) P µζ +1 (cosh ξ ) − ( ζ + 1) cosh ξP µζ (cosh ξ ) i , (4.26)for α ( ξ ) = sinh ξP µζ (cosh ξ ) we obtain (cid:2) sinh ξ∂ ξ + sinh ξ cosh ξ∂ ξ (cid:3) h sinh ξP µζ (cosh ξ ) i = sinh ξ h ζ ( ζ + 1) sinh ξ + µ + 1 + 2 sinh ξ − ζ + 1)(1 + sinh ξ ) i P µζ (cosh ξ ) + 2 sinh ξ cosh ξ ( ζ − µ + 1) P µζ +1 (cosh ξ ) . (4.27)Thus finally, for the comoving time γ ( ξ ) we obtain2 γ ( ξ )( − k ) / sinh ξ cosh ξ = − κ h ζ ( ζ + 1) sinh ξ + µ + 1 + 2 sinh ξ − ζ + 1)(1 + sinh ξ ) (cid:21) P µζ (cosh ξ ) − ( κ/ ν + 1) P µζ (cosh ξ ) − κ cosh ξ ( ζ − µ + 1) P µζ +1 (cosh ξ ) − sinh ξP µζ (cosh ξ ) , (4.28)where µ and ζ are given in (4.19). V. VECTOR SECTOR SOLUTION FROM RECOMBINATION UNTIL THE CURRENT ERA
As we noted above, in the vector sector we need to find solutions to the conformal time (3.17), viz. η ( ∂ τ + 2 k − ˜ ∇ a ˜ ∇ a )( B i − ∂ τ E i ) + Ω ( B i − ∂ τ E i ) = 0 . (5.1)Given that Ω = − ( − k/σ ) / / sinh(( − k ) / τ ), on changing the variable to ρ = ( − k ) / τ , on setting B i − ∂ τ E i = D i ,and on introducing a separation constant according to ( ˜ ∇ a ˜ ∇ a + ( − k ) A V ) D i = 0, we rewrite (5.1) as ∂ D i ∂τ + ( A V − D i + D i κ sinh ρ = 0 , (5.2)where as before κ = ση . Then on setting D i = sinh ρC i we obtain¨ C i + 2 cosh ρ sinh ρ ˙ C i + ( A V − C i + C i κ sinh ρ = 0 , (5.3)where the dot denotes the derivative with respect to ρ . We recognize this equation as (4.5), and set ν = A V − ζ = − ± (2 − A V ) / = − ± iν, µ = 14 − κ , (5.4)the conformal time dependence is given by B i ( ρ ) − ∂ τ E i ( ρ ) = ǫ i sinh / ρP µζ (cosh ρ ) = ǫ i sinh / ρ − µ ) coth µ ( ρ/ F ( − ζ, ζ + 1; 1 − µ ; − sinh ( ρ/ , (5.5)where ǫ i is a transverse polarization vector.6 A. Comoving Time Vector Sector Solutions
Converting (5.1) to comoving time where now D i = B i − a ( t )( ∂E i /∂t ) gives η (( − k ) A V + a ∂ t + a ( ∂ t a ) ∂ t + 2 k ) D i + a D i = 0 . (5.6)On changing the variable to ξ = σ / t we obtain¨ D i + ˙ aa ˙ D i + ( − k ) A V + 2 kσa D i + D i ση = 0 , (5.7)where the dot denotes the derivative with respect to ξ . Setting D i = σ / ( − k ) − / a / F i gives¨ F i + 2 ˙ aa ˙ F i + ( − k ) A V + 2 kσa F i + 1 ση F i + ˙ a a F i + ¨ a a F i = 0 . (5.8)Setting κ = ση and a = ( − k/σ ) / sinh ξ , so that D i = sinh / ξF i , yields¨ F i + 2 cosh ξ sinh ξ ˙ F i + A V − / ξ F i + (cid:18) κ + 34 (cid:19) F i = 0 . (5.9)We now proceed as in (4.17) only with the replacement A S → A V − , (5.10)so that now with A V − ν we have ζ = − ± (cid:18) − κ (cid:19) / , µ = 2 − A V = − ν . (5.11)For the comoving time dependence we thus have B i ( ξ ) − a ( t ) ∂ t E i ( ξ ) = ǫ i P µζ (cosh ξ ) = ǫ i − µ ) coth µ ( ξ/ F ( − ζ, ζ + 1; 1 − µ ; − sinh ( ξ/ , (5.12)where ǫ i is a transverse polarization vector. VI. TENSOR SECTOR SOLUTION FROM RECOMBINATION UNTIL THE CURRENT ERA
In the tensor sector we have to solve the conformal time (3.18). On setting ρ = ( − k ) / τ and introducing a tensorsector separation constant that obeys [ ˜ ∇ a ˜ ∇ a + ( − k ) A T ] E ij = 0, after separating the variables (3.18) takes the form − kη Ω − (cid:2) ( ∂ ρ − A T ) − ∂ ρ (cid:3) E ij ( ρ ) = − (cid:2) ∂ ρ − − ( ∂ ρ Ω) ∂ ρ + A T (cid:3) E ij ( ρ ) . (6.1)On setting Ω = − ( − k/σ ) / / sinh ρ and κ = ση (6.1) takes the form κ sinh ρ (cid:2) ( ∂ ρ − A T ) − ∂ ρ (cid:3) E ij ( ρ ) = − (cid:20) ∂ ρ − − ρ sinh ρ ∂ ρ + A T (cid:21) E ij ( ρ ) . (6.2)Symbolically we write (6.2) as Y = − X . On setting E ij ( ρ ) = F ij ( ρ ) sinh ρ and dropping the ( i, j ) indices, on writing s = sinh ρ , c = cosh ρ , we find that following some algebra X and Y are given by X = s ¨ F + 2 cs ˙ F − F − s F + A T s F, (6.3)and Y = κ [ s .... F + 8 s c ... F + (12 s + 16 s ) ¨ F − (8 s + 12 s ) F + 2 A T s ¨ F + 8 A T s c ˙ F + A T (4 s + 4 s ) F + A T s F ]= κ [ s ¨ X + 2 sc ˙ X + A T s X − s X ] , (6.4)where the dot denotes the derivative with respect to ρ . We thus find that Y can be expressed as a derivative of X ,with (6.2) then factorizing into the remarkably compact form (cid:2) κ [ s ∂ ρ + 2 sc∂ ρ + A T s − s ] + 1 (cid:3) X = 0 . (6.5)7 A. Conformal Time Tensor Sector Solutions
If we define D = sinh ρ∂ ρ + 2 sinh ρ cosh ρ∂ ρ + A T sinh ρ − ρ − , (6.6)we can then write (6.5) as [ κ ( D + 2) + 1] DF = 0 . (6.7)To solve (6.7) we identify two classes of solutions: DF = 0 , [ κ ( D + 2) + 1] DF = 0 . (6.8)The first solution obeys: (cid:20) ∂ ρ + 2 cosh ρ sinh ρ ∂ ρ + A T − − ρ (cid:21) F = 0 . (6.9)Comparing with (4.5) we replace A S by A T − /κ by −
2. Thus on setting ν = A T −
3, from the conformaltime (4.10) we obtain ζ = − ± (3 − A T ) / = − ± iν, µ = 94 . (6.10)Consequently, with µ = 3 / F is given by F = (sinh ρ ) − / P / − / ± iν (cosh ρ ) = − ν cos νρ sinh ρ + ν sin νρ cosh ρ sinh ρ . (6.11)For µ = 3 / P / − / ± iν (cosh ρ ). It may be found in [29, 30, 68] and (7.4) below, and we haveexhibited it in (6.11).For the second solution we set X = DF and rewrite (6.8) as[ D + 2 + 1 /κ ] X = 0 , D [ X + (2 + 1 /κ ) F ] = 0 , (6.12)i.e., (cid:20) ∂ ρ + 2 cosh ρ sinh ρ ∂ ρ + A T − κ sinh ρ (cid:21) X = 0 , (6.13) (cid:20) ∂ ρ + 2 cosh ρ sinh ρ ∂ ρ + A T − − ρ (cid:21) [ X + (2 + 1 /κ ) F ] = 0 . (6.14)We recognize (6.13) as (4.5) with A T − A S and can thus set ζ = − ± (3 − A T ) / = − ± iν, µ = 14 − κ ,X = (sinh ρ ) − / P (1 / − /κ ) / − / ± iν (cosh ρ ) . (6.15)On comparing with (4.5), for (6.14) we have A S = A T −
2, 1 /κ = −
2. Thus we obtain ζ = − ± (3 − A T ) / = − ± iν, µ = 94 ,X + (2 + 1 /κ ) F = (sinh ρ ) − / P / − / ± iν (cosh ρ ) , (6.16)We recognize as X + (2 + 1 /κ ) F as F . From (6.16) we obtain(2 + 1 /κ ) F = (sinh ρ ) − / P / − / ± iν (cosh ρ ) − (sinh ρ ) − / P (1 / − /κ ) / − / ± iν (cosh ρ ) . (6.17)Given F and F , the two classes of solutions depend on the conformal time as E ij ( ρ ) = A ij sinh / ρP / − / ± iν (cosh ρ ) + B ij sinh / ρP (1 / − /κ ) / − / ± iν (cosh ρ ) , (6.18)where A ij and B ij are transverse-traceless polarization tensors.8 B. Comoving Time Tensor Sector Solutions
To convert to comoving time we rewrite (6.9), (6.13) and (6.14) as (cid:20) ∂ ρ − ∂ ρ ΩΩ ∂ ρ + A T − − σ Ω ( − k ) (cid:21) F = 0 , (6.19) (cid:20) ∂ ρ − ∂ ρ ΩΩ ∂ ρ + A T − σ Ω ( − k ) κ (cid:21) X = 0 , (6.20) (cid:20) ∂ ρ − ∂ ρ ΩΩ ∂ ρ + A T − − σ Ω ( − k ) (cid:21) [ X + (2 + 1 /κ ) F ] = 0 . (6.21)Thus in comoving time with ξ = σ / t , a ( t ) = ( − k/σ ) / sinh ξ and d/dρ = sinh ξd/dξ we have (cid:20) ∂ ξ − cosh ξ sinh ξ ∂ ξ + A T − ξ − (cid:21) F = 0 , (6.22) (cid:20) ∂ ξ − cosh ξ sinh ξ ∂ ξ + A T − ξ + 1 κ (cid:21) X = 0 , (6.23) (cid:20) ∂ ξ − cosh ξ sinh ξ ∂ ξ + A T − ξ − (cid:21) [ X + (2 + 1 /κ ) F ] = 0 . (6.24)Setting F = sinh / ξβ , X = sinh / ξβ , X + (2 + 1 /κ ) F = sinh / ξβ , ν = A T − (cid:20) ∂ ξ + 2 cosh ξ sinh ξ ∂ ξ + ν + 1 / ξ − (cid:21) β = 0 , (6.25) (cid:20) ∂ ξ + 2 cosh ξ sinh ξ ∂ ξ + ν + 1 / ξ + 1 κ + 34 (cid:21) β = 0 , (6.26) (cid:20) ∂ ξ + 2 cosh ξ sinh ξ ∂ ξ + ν + 1 / ξ − (cid:21) β = 0 . (6.27)Comparing with (4.5) and (4.10) for β and β we obtain1 κ → ν + 14 , A S → − , ζ = − ± , µ = − ν , (6.28)and for β we obtain 1 κ → ν + 14 , A S → κ + 34 , ζ = − ± (cid:18) − κ (cid:19) / , µ = − ν . (6.29)Comparing with (4.13), a solution to (4.5) in the coordinates that appear in (4.5), each β is given bysinh − / ξP µζ (cosh ξ ) with appropriate µ and ζ . Thus each F i is given by sinh ξP µζ (cosh ξ ). To convert from F and F to E ij we note that in conformal time we set E ij = F ij sinh ρ . With Ω( ρ ) = ( − k/σ ) / / sinh ρ and a ( t ) = ( − k/σ ) / sinh ξ , we set E ij = F ij ( − k/σ ) / Ω ( ρ ) = F ij ( − k/σ ) /a ( ξ ) = F ij / sinh ξ . Finally then, for E ij the two classes of solutions depend on the comoving time as E ij ( ξ ) = A ij sinh ξ P ± iνζ (cosh ξ ) + B ij sinh ξ P ± iνζ (cosh ξ ) , (6.30)9where A ij and B ij are transverse-traceless polarization tensors.Finally, as a check on all of the comoving time solutions, we compare their recombination era behavior with therecombination era behavior that had been determined in [30]. To this end we note with the recombination era beingsuch that ξ = σ / t is small enough that we can set sinh ξ = ξ , to lowest order in ξ we can set F ( − ζ, ζ + 1; 1 − µ ; − sinh ( ξ/ ∼ ζ ( ζ + 1)(1 − µ ) − sinh ( ξ/ ∼
1. Thus from (4.20), (5.12) and (6.30) we obtain α ∼ t ± iν , B i − a ( t )( ∂E i /∂t ) ∼ t ± iν , E ij ∼ t ± iν − at small t , just as had been found at recombination in [30]. We thus establishthat with real ν the solutions oscillate at small time [69].Also we note that for the scalar, vector and tensor solutions we had set A S − ν , A V − ν , A T − ν , withour intent here being that all solutions be associated with one and the same ν . By study of the spatial behavior ofthe solutions we now show that this is in fact the case. Moreover, as the small t solutions already show, for solutionsthat oscillate in time we need ν to be real, and thus need the separation constants to obey A S ≥ A V ≥ A T ≥ ν being the k < q + q + q ) / in a flat three-space. VII. THE SPATIAL STRUCTURE OF THE SOLUTIONS
To complete the characterization of the fluctuation solutions we need to specify the spatial behavior. Since thishad already been discussed in [29, 30] we briefly state the results. To analyze the spatial behavior of the modes it isconvenient to set r = sinh χ/ ( − k ) / , τ = ρ/ ( − k ) / , so that the background metric takes the form ds = ( − k ) − Ω ( τ ) (cid:2) dρ − dχ − sinh χdθ − sinh χ sin θdφ (cid:3) . (7.1) A. Scalar Sector Spatial Structure
For the scalar fluctuations described by a generic scalar S we need to solve (cid:16) ˜ ∇ a ˜ ∇ a + ( − k ) A S (cid:17) S = 0 , (7.2)where ( − k ) A S is the separation constant we introduced earlier when discussing the temporal behavior of the solutions.On setting S ( χ, θ, φ ) = S ℓ ( χ ) Y mℓ ( θ, φ ) (7.2) reduces to (cid:20) d dχ + 2 cosh χ sinh χ ddχ − ℓ ( ℓ + 1)sinh χ + A S (cid:21) S ℓ ( χ ) = 0 . (7.3)On making the identification ℓ ( ℓ +1) = − /κ , we recognize this equation as being none other than the same fluctuationequation (4.5) that had met earlier in studying the time behavior. Now while we could identify the solution as P µζ (cosh χ ) with ζ = − / ± iν , µ = ( ℓ + 1 / , because ℓ is an integer (7.3) can be solved directly in terms ofelementary functions. Specifically, there are two classes of solutions to this second-order differential equation, labelledˆ S ℓ and ˆ f ℓ ˆ S ℓ , and they are of the formˆ S ℓ = sinh ℓ χ (cid:18) χ ddχ (cid:19) ℓ +1 cos νχ, ˆ f ℓ ˆ S ℓ = ˆ S ℓ Z dχ sinh χ ˆ S ℓ , ν = A S − , (7.4)with both solutions being even in ν . The ˆ S ℓ solution is given in [29, 30, 68] and the ˆ f ℓ ˆ S ℓ solution is given in [29, 30].Both of these sets of solutions with real and continuous ν (i.e., real and continuous A S >
1) are complete. With theasymptotic behavior of the solutions being of the form e − χ e ± iνχ [29, 30], the solutions indeed oscillate if ν is real.Moreover, at χ = 0 the solutions behave as χ ℓ or as χ − ℓ − , with there always being one solution that is bounded as e − χ e ± iνχ at χ = ∞ and well behaved at χ = 0 for all ℓ , and one solution that is bounded as e − χ e ± iνχ at χ = ∞ butbadly behaved at χ = 0 for all ℓ . Typical examples are ˆ S ℓ =0 = − ν sin νχ/ sinh χ , ˆ f ℓ =0 ˆ S ℓ =0 = cos νχ/ν sinh χ . Bothsolutions are bounded at χ = ∞ , with the first solution behaving as χ at χ = 0 and the other as χ − .In [68] a master equation was introduced that provides normalization factors for scalar sector modes that arepropagating in a space with F spatial dimensions. We introduce Z ν,β ( χ ) = A ( ν, β, F ) sinh β χ (cid:18) χ ddχ (cid:19) ( F − / β cos νχ, (7.5)0with ℓ = β + ( F − / A ( ν, β, F ) = 2 / [ πν ( ν + 1 )( ν + 2 ) ...... ( ν + (( F − / β ) )] / . (7.6)With the χ dependence of F -dimensional integration measure being sinh F − χ , the modes obey the Dirac deltafunction orthonormality condition [68] Z ∞ dχ sinh χ sinh ( F − / χZ ν ,β ( χ ) sinh ( F − / χZ ∗ ν ,β ( χ ) = δ β ,β δ ( ν − ν ) . (7.7)Thus for the dimension F = 3 that we are interested in we have Z ∞ dχ sinh χZ ν ,ℓ ( χ ) Z ∗ ν ,ℓ ( χ ) = δ ℓ ,ℓ δ ( ν − ν ) . (7.8)In regards to this normalization condition we note that since maximally symmetric three-spaces with metric dχ + sinh χdθ + sinh χ sin θdφ and integration measure sinh χ sin θ are conformal to flat, the delta functionnormalization that is needed for plane waves propagating in a flat space translates into the delta function normal-ization for the Z ν,β ( χ ) that is given here, with the continuous parameter ν replacing the familiar continuous linearmomentum variable. B. Vector Sector Spatial Structure
For the vector fluctuations described by a generic vector V i we need to solve (cid:16) ˜ ∇ a ˜ ∇ a + ( − k ) A V (cid:17) V i = 0 , (7.9)where ( − k ) A V is the separation constant we introduced earlier. While the various components of V i are mixed in(7.9), as shown in [29], for a transverse V i that obeys ˜ ∇ i V i = 0 the equation for V only involves V (i.e., V χ ). Andon setting V ( χ, θ, φ ) = ˆ V ℓ ( χ ) Y mℓ ( θ, φ ) the equation for V reduces to (cid:20) d dχ + 4 cosh χ sinh χ ddχ + 2 + A V + 2sinh χ − ℓ ( ℓ + 1)sinh χ (cid:21) ˆ V ℓ ( χ ) = 0 . (7.10)On setting ˆ V ℓ = α ℓ / sinh χ we find that (7.10) takes the form1 L (cid:20) d dχ + 2 cosh χ sinh χ ddχ − ℓ ( ℓ + 1)sinh χ + A V − (cid:21) α ℓ = 0 . (7.11)We recognize (7.11) as being in the same form as (7.3), but with A S replaced by A V − ν = A V − e − χ e ± iνχ , and at χ = 0 as χ ℓ − , χ − ℓ − . Thus the solutions indeedoscillate if ν is real, and with real and continuous ν (i.e., real and continuous A V >
2) the solutions are complete.Moreover, at χ = 0 there is always one solution that is bounded as e − χ e ± iνχ at χ = ∞ and well behaved at χ = 0 for all ℓ ≥
1, and one solution that is bounded as e − χ e ± iνχ at χ = ∞ but badly behaved at χ = 0 for all ℓ ≥
1. A typical example of a solution that behaves as e − χ e ± iνχ asymptotically and is well-behaved at χ = 0 isˆ V ℓ =1 = ν sin νχ cosh χ/ sinh χ − ν cos νχ/ sinh χ .For the normalization of the vector modes we recall that while the normalization condition given in (7.7) wasestablished for scalar modes in spaces with dimension F , it was noted in [30] that these same conditions hold forvector modes in three dimensions if we set F = 5, ℓ = β + 1 in (7.7). Thus for vector modes in three spatial dimensionswe obtain Z ∞ dχ sinh χ sinh χZ ν ,ℓ − ( χ ) sinh χZ ∗ ν ,ℓ − ( χ ) = δ ℓ ,ℓ δ ( ν − ν ) . (7.12)As we see, compared with the scalar (7.8) the vector sector measure includes two extra factors of sinh χ in order tobalance the replacement of ˆ V ℓ by α ℓ / sinh χ , so that asymptotically for a scalar that behaves as e − χ e ± iν the vectorbehaves as e − χ e ± iν . Moreover, the change in β by one unit shifts ℓ by one unit so that only ℓ ≥ χ = 0.Using these same techniques and the formalism presented in [29, 30] we could solve for V and V as well, but donot actually need to do so explicitly since for study of the anisotropy in the cosmic microwave background only lineof sight radial mode fluctuations are detectable by a current era observer (see e.g. [3]).1 C. Tensor Sector Spatial Structure
For the tensor fluctuations described by a generic tensor T ij we need to solve (cid:16) ˜ ∇ a ˜ ∇ a + ( − k ) A T (cid:17) T ij = 0 , (7.13)where ( − k ) A T is the separation constant we introduced earlier. While the various components of T ij are mixed in(7.13), as shown in [29] for a transverse-traceless T ij the equation for T only involves T (viz. T χ,χ ). And on setting T ( χ, θ, φ ) = ˆ T ℓ ( χ ) Y mℓ ( θ, φ ) the equation for T reduces to (cid:20) d dχ + 6 cosh χ sinh χ ddχ + 6 + 6sinh χ − ℓ ( ℓ + 1)sinh χ + A T (cid:21) ˆ T ℓ ( χ ) = 0 . (7.14)On setting ˆ T ℓ = γ ℓ / sinh χ we find that (7.14) takes the form (cid:20) d dχ + 2 cosh χ sinh χ ddχ − ℓ ( ℓ + 1)sinh χ − A T (cid:21) γ ℓ = 0 . (7.15)We recognize (7.15) as being in the same form as (7.3), but with A S replaced by A T − ν = A T − e − χ e ± iνχ , and at χ = 0 as χ ℓ − , χ − ℓ − . Thus the solutions indeedoscillate if ν is real, and with real and continuous ν (i.e., real and continuous A T >
3) the solutions are complete.Moreover, at χ = 0 there will always be one solution that is bounded as e − χ e ± iνχ at χ = ∞ and well behaved at χ = 0 for all ℓ ≥
2, and one solution that is bounded as e − χ e ± iνχ at χ = ∞ but badly behaved at χ = 0 for all ℓ ≥
2. A typical example of a solution that behaves as e − χ e ± iνχ asymptotically and is well-behaved at χ = 0 isˆ T ℓ =2 = 3 ν cos νχ cosh χ/ sinh χ − ν (2 − ν ) sin νχ/ sinh χ − ν sin νχ/ sinh χ .For the normalization of the tensor modes we recall that while the normalization condition given in (7.7) wasestablished for scalar modes in spaces with dimension F , it was noted in [30] that these same conditions hold fortensor modes in three dimensions if we set F = 7, ℓ = β + 2 in (7.7). Thus for tensor modes in three spatial dimensionswe obtain Z ∞ dχ sinh χ sinh χZ ν ,ℓ − ( χ ) sinh χZ ∗ ν ,ℓ − ( χ ) = δ ℓ ,ℓ δ ( ν − ν ) . (7.16)As we see, compared with the scalar (7.8) the tensor sector measure includes two extra factors of sinh χ in order tobalance the replacement of T ,ℓ by γ ℓ / sinh χ , so that asymptotically for a scalar that behaves as e − χ e ± iν the tensorbehaves as e − χ e ± iν . Moreover, the change in β by two units shifts ℓ by two units so that only ℓ ≥ χ = 0.Using these same techniques and the formalism presented in [29, 30] we could solve for the other components of T ij as well, but as with the vector fluctuations, we do not actually need to do so explicitly since only line of sight radialmode fluctuations in the cosmic microwave background are detectable by a current era observer. VIII. THE FULL SOLUTION FROM RECOMBINATION TO THE CURRENT ERA
Putting everything together, in conformal cosmology the full comoving time radial mode solutions from recombi-nation to the current era are of the form α = A ( ν, ℓ,
3) ˆ S ℓ ( χ ) Y mℓ ( θ, φ ) sinh( σ / t ) P µζ (cosh( σ / t )) , ζ = − ± (cid:18) − ση (cid:19) / , µ = 1 − A S = − ν , (8.1)2 dadt γ = η a (cid:18) ˜ ∇ a ˜ ∇ a − a ∂ ∂t − a dadt ∂∂t (cid:19) α − α, (8.2) B − a ( t ) ∂ t E = ǫ A ( ν, ℓ − ,
5) ˆ V ℓ ( χ ) Y mℓ ( θ, φ ) P µζ (cosh( σ / t )) , ζ = − ± (cid:18) − ση (cid:19) / , µ = 2 − A V = − ν , (8.3)2 E = A ( ν, ℓ − ,
7) ˆ T ℓ ( χ ) Y mℓ ( θ, φ ) 1sinh( σ / t ) (cid:20) A P µζ (cosh( σ / t ) + B P µζ (cosh( σ / t ) (cid:21) ,ζ = − ± , ζ = − ± (cid:18) − ση (cid:19) / , µ = 3 − A T = − ν , (8.4)for all real and positive ν . In the conformal theory these radial scalar, vector and tensor mode solutions are exactwithout approximation from recombination until the current era. Acknowledgments
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2) it follows that P ζ ( z ) is real if z is real and ζ is in the conical function form ζ = − / iλ where λ is real. Also, P − / iλ ( z ) with real z has an infinite number of real zeros when z > ξ since z = cosh ξ , with oscillations that are damped. Moreover,numerically we have found in some typical cases that RE[ P iν − / iλ ( z )] with real nonzero ν and real z also has oscillationsthat are damped. For confirmation, we note that at large z P iνζ ( z ) is known to behave as z ζ ζ Γ[ ζ +1 / / (Γ[1 / ζ − iν +1])for any ζ and ν . Thus with ζ = − / ± (1 / − /κ ) / as given (4.19), (5.11), (6.29), the scalar, vector and tensor comovingtime solutions will only oscillate at late times if 1 /κ > /
4, i.e., if 0 < κ <
4. Since κ = ση and since σ is positive,this can only occur if η is positive, just as we noted is in fact the case [22, 64]. And with κ < λα g <
12. (This result replaces the result presented in [30], where it was thought that one getsoscillating solutions for any positive ηη