Cosmological Perturbations without Inflation
aa r X i v : . [ a s t r o - ph . C O ] N ov Cosmological Perturbations without Inflation
Fulvio Melia ‡ Department of Physics, The Applied Math Program, and Department ofAstronomy,The University of Arizona, AZ 85721, USA
Abstract.
A particularly attractive feature of inflation is that quantumfluctuations in the inflaton field may have seeded inhomogeneities in the cosmicmicrowave background (CMB) and the formation of large-scale structure. Inthis paper, we demonstrate that a scalar field with zero active mass, i.e.,with an equation of state ρ + 3 p = 0, where ρ and p are its energy densityand pressure, respectively, could also have produced an essentially scale-freefluctuation spectrum, though without inflation. This alternative mechanism isbased on the Hollands-Wald concept of a minimum wavelength for the emergenceof quantum fluctuations into the semi-classical universe. A cosmology with zeroactive mass does not have a horizon problem, so it does not need inflation tosolve this particular (non) issue. In this picture, the 1 ◦ − ◦ fluctuations inthe CMB correspond almost exactly to the Planck length at the Planck time,firmly supporting the view that CMB observations may already be probing trans-Planckian physics.PACS number: 04.20.Ex, 95.36.+x, 98.80.-k, 98.80.Jk
1. Introduction
In spite of its many successes, standard big-bang cosmology suffers from severalconceptual and physical anomalies and inexplicable observational puzzles, such as the‘horizon’ problem, in which the cosmic microwave background (CMB) temperatureis relatively uniform everywhere, even though causally connected regions at lastscattering are much smaller than the horizon size today. Founded on a combinationof classical and quantum physical principles [1, 2, 3], the inflationary paradigm wasdeveloped to address these issues [4, 5, 6, 7, 8]. But though the idea of inflation isvery flexible, it has yet to find expression in a comprehensive, self-consistent modelthat accounts for all of the observations [9].Many variations of the inflationary concept exist today [2, 3, 10, 11, 12, 13, 14, 15],allowing us to parametrize features in the early universe, but none may represent afinal comprehensive answer if their fundamentally semi-classical nature is at odds withPlanckian (or even pre-Planckian) scale physics. On the other hand, an inflationaryphase may have been associated with a scalar (‘inflaton’) field beyond the Planckscale, whose identity could eventually be revealed via extensions to the StandardModel based on supergravity, grand unified theories, or even string theory.Most of the problems arising in standard big-bang cosmology are due to theinevitable decelerated expansion associated with a radiation or matter dominated ‡ John Woodruff Simpson Fellow. cosmic fluid. Inflation circumvents this deficiency by postulating the existence ofan early phase of accelerated expansion, during which proper distances grew fasterthan the gravitational horizon size, which in some inflationary scenarios actuallydid not grow at all (or grew very slowly) during this brief period. Thus, physicaldistances would have been pushed beyond the Hubble radius, which potentially solvesthe horizon problem.An additional attractive feature of inflation—central to the subject of thispaper—is that quantum fluctuations of the inflaton field may have generated densityperturbations seeding the formation of large-scale structure [16, 17, 18, 19]. Thesefluctuations would have been stretched on large scales by the brief acceleratedexpansion and, in the simplest version of the single inflaton field scenario, would havebecome ‘frozen’ after their wavelength exceeded the Hubble radius. And since inflationmust have somehow ended in order for the Universe to subsequently re-establish itsradiation and matter dominated expansion, the perturbations would have crossed backinside the Hubble radius.In this picture, fluctuations with ever larger co-moving wavenumbers crossedthe horizon and became frozen at progressively later times, a process naturallyproducing nearly scale-invariant spectra. This expectation has been largely confirmedby measurements of the temperature anisotropies in the CMB, generating considerableenthusiasm for the inflationary paradigm, well beyond its early success in apparentlyresolving other long-standing issues, such as the aforementioned horizon problem.Even so, tension continues to grow between the overall expectations of theinflationary model and some key observations including, and especially, of the CMBanisotropies. The emergence of greater detail in all-sky maps has revealed severalunexpected features on large scales, first reported by the Cosmic Background Explorer(COBE) Differential Microwave Radiometer (DMR) collaboration [20]. Disagreementwith theory arises from an apparent alignment of the largest modes of CMB anisotropy,as well as the absence of any angular correlation at angles greater than ∼ ◦ .The latter is particularly troublesome because all fluctuations presumably exited thehorizon during inflation, which should have produced a correlation at all angles. Theseunexpected features have been explained as possibly due to cosmic variance withinthe standard model [21], but this explanation may not be completely satisfactory.An analysis of the differences between the observed angular correlation functionand that predicted by inflation in ΛCDM [22] has revealed that only ∼ .
03% ofΛCDM model CMB skies have a variance larger than that of the sky observed withthe Wilkinson Microwave Anisotropy Probe (WMAP) [21]. We may simply be dealingwith foreground subtraction issues. But there are indications that the differencesbetween theory and observations may be due to more than just randomness. Forexample, the well-defined shape of the observed angular correlation function, witha minimum at ∼ ◦ , is at odds with the expectation that the data points wouldnot have lined up as they do within the variance window if stochastic processes weresolely to blame. More importantly, the observed angular correlation goes to zerobeyond ∼ ◦ . While variance could have resulted in a function with a different slopethan that predicted by inflation, it seems unlikely that this randomly generated slopewould be close to zero above ∼ ◦ .This tension has been exacerbated by the more recent Planck results [23]. Theprobability of the
Planck sky being consistent with inflation in ΛCDM is ∼ . Planck ).The apparent lack of temperature correlations at large angles is robust and increases instatistical significance as the quality of the measurements improves, suggesting thatinstrumental issues are not the cause. Indeed, if it turns out that the absence oflarge-angle correlation is real, this may be the most significant outcome of the CMBobservations, because it would essentially invalidate any role that inflation might haveplayed in the universal expansion.In this paper, we consider an alternative scenario for generating quantumfluctuations in the early universe, not solely because of the potential problems facinginflation in accounting for all the data, but more so because another Friedmann-Robertson-Walker (FRW) cosmology, known as the R h = ct universe [25, 26] (see alsoRef. [27] for a non-technical introduction) has been shown in recent years to accountfor a broad range of high-precision cosmological measurements better (in some cases,significantly better) than ΛCDM, the current standard model based on inflation.For example, whereas the inflationary paradigm has trouble explaining theabsence of any angular correlation in the CMB beyond ∼ ◦ , this characteristicsimply results from the size of the gravitational horizon (i.e., the Hubble radius) atlast scattering in the R h = ct universe [28]. A rather compelling example of howthe predictions of ΛCDM and R h = ct differ in their comparisons with the datais provided by a recent application of the Alcock-Paczy´nski test [29], based on thechanging ratio of angular to spatial/redshift size of (presumed) spherically-symmetricsource distributions with distance, to the most accurate measurements of the baryonacoustic oscillation (BAO) scale. The use of this diagnostic with newly acquireddata on the anisotropic distribution of the BAO peaks from SDSS-III/BOSS-DR11 ataverage redshifts h z i = 0 .
57 [30] and h z i = 2 .
34 [31], disfavors the current concordance(ΛCDM) model at better than a 99 .
34% CL, while the probability that the R h = ct universe is consistent with these data is ∼ .
96, i.e., essentially one [32].The question concerning how quantum fluctuations were generated in the earlyuniverse is critical to this whole discussion because, whereas ΛCDM probably cannotsurvive without inflation, the R h = ct universe does not have or need it. Thiscosmology did not undergo a period of decelerated expansion, and therefore avoidsthe horizon problem altogether [33]. So while this paper is in principle motivatedseparately by (1) a desire to alleviate the growing tension between the predictionsof inflation and the ever improving observations, and (2) a need to strengthen theviability of the R h = ct universe by uncovering a mechanism to generate cosmologicalperturbations in this model, in reality these two goals overlap considerably. Ourprincipal task is to determine how and why quantum fluctuations could have grownin R h = ct without inflation.Before we begin our development of this mechanism, however, it is worthwhileconsidering several earlier attempts at producing quantum fluctuations withoutinflation, and how they differ from the proposal we are making here. In their work,Bengochea et al. [34] adopted the Hollands-Wand concept, but focused primarilyon the question of how classicalization may actually occur in such a scenario. Thisissue, of how a homogeneous and isotropic quantum fluctuation is converted intoactual inhomogeneities and anisotropies at the classical scale, is common to all modelsinvoking a quantum origin for the perturbations, and is yet to be resolved (see alsoref. [35]). In their analysis, these authors adopted a standard cosmological background(other than inflation), whereas in this paper we will focus exclusively on the zero activemass equation-of-state associated with the R h = ct model. The manner in whichthe modes are born and subsequently stretch and grow is quite different in these twocases. As we shall see, the mode wavelength grows as a constant fraction of the Hubbleradius, so its transition from the Planck domain to the ∼ ◦ − ◦ scale associatedwith the CMB is smooth and does not involve multiple steps, such as one encountersduring inflation, where modes cross and re-cross the horizon during their evolution.Nonetheless, this paper will not be fully addressing the question of classicalization,which remains a largely unresolved problem.A non-inflationary mechanism for generating the perturbation spectrum hasalso been considered in ekpyrotic [36] and cyclic [37] models. Here too, however,these models have the common feature that the perturbations originated asquantum fluctuations which exited and re-entered the horizon during their evolution.Interestingly, this process occurs for both expanding cosmologies (e.g., in the standardmodel) and a contracting universe, with an appropriate alteration to the evolution inthe scale factor a ( t ) [38]. The cyclic model repeats its periods of expansion andcontraction, the latter of which is identical to the ekpyrotic case. The R h = ct modelthat we focus on in this paper is unique, in that this is the only case in which allproper distances and the Hubble radius expand at the same rate. As we shall see,the mechanism for producing a near-scale free spectrum is therefore simpler, withthe added advantage that the observed scale of fluctuations in the CMB traces backdirectly to the Planck wavelength at the Planck time. None of the other models havethis feature, which provides some justification for the argument that perturbationswere essentially trans-Plancking in nature.In § II of this paper, we briefly summarize the origin and essential characteristicsof the R h = ct universe, and then discuss the cosmological dynamics in this model in § III. The cosmological perturbations are introduced in § IV, where we describe someof this model’s most significant predictions. We end with our conclusions in § V.
2. The R h = ct Universe
The R h = ct universe is an FRW cosmology in which the underlying symmetries ofthe metric, with particular reference to Weyl’s postulate [39], are used to incorporatethe influence of a gravitational horizon on the expansion dynamics [25, 26, 40]. Themodel is based on standard general relativity (GR), and the Cosmological principle isadopted from the start, just like any other FRW cosmology, but it equally addressesthe consequences of Weyl’s postulate, whose role in shaping the FRW metric is oftenignored.It is commonly assumed that Weyl’s postulate is already incorporated intoall forms of the FRW metric, and is therefore given far less attention than theCosmological principle. Simply stated, Weyl’s postulate holds that any proper distance R ( t ) is a product of a universal expansion factor a ( t ) (dependent only on cosmic time t ) and an unchanging co-moving radius r : R ( t ) = a ( t ) r . We conventionally write theFRW metric adopting this coordinate definition, along with t , which is actually theobserver’s proper time in his/her free-falling frame.But its impact is far greater than this. Consider, for instance, the Misner-Sharpmass M —defined in terms of the proper mass density ρ/c and proper volume 4 πR / proper mass M , the gravitational radiusof the Universe is R h ≡ GM/c [25, 40], which actually coincides with the betterknown Hubble radius c/H ( t ). Given its definition, R h (and therefore the Hubbleradius) is a proper distance [40], so this radius must comply with Weyl’s postulate,the consequence of which is the unique choice a ( t ) = ( t/t ) for the expansion factor,where t is the current age of the Universe [26]. Those familiar with the propertiesof the Schwarzschild or Kerr metrics are not at all surprised by this constraint, whichleads to the result that the gravitational radius must be receding from us at speed c —hence the name ‘ R h = ct ’ for this model. The Hubble radius was in fact defined tobe the distance at which the Hubble speed equals c even before it was recognized asanother manifestation of the gravitational horizon. In black-hole spacetimes, a free-falling observer sees the event horizon approaching them at speed c , so this propertyof R h = ct is quite familiar in the context of standard GR.One of the principal differences between R h = ct and other FRW cosmologies,such as ΛCDM, is how they handle the energy density ρ and pressure p , and theirtemporal evolution. In ΛCDM we routinely start with the constituents in the cosmicfluid, and assume their equations-of-state, and then solve the dynamics equations todetermine the expansion rate as a function of time. In R h = ct , on the other hand,the symmetries of the FRW metric and the properties of the gravitational horizon,uniquely specify the spacetime curvature, and hence the expansion rate, strictly fromjust the value of the total energy density ρ , without us having to know the specificsof the constituents themselves. In this model, the constituents of the Universe mustpartition themselves in such a way as to satisfy the constant expansion rate required bythe R h = ct condition. Insofar as the dynamics is concerned, all that matters is ρ andthe overall equation of state p ≡ wρ . So while one assumes ρ = ρ m + ρ r + ρ Λ in ΛCDM,i.e., that the principal constituents are matter, radiation, and a cosmological constantΛ, and then infers w from the equations-of-state assigned to them, in R h = ct , it isthe aforementioned symmetries and other constraints from GR that force the R h = ct universe to have the unique equation-of-state [25, 26] ρ + 3 p = 0 . (1)The R h = ct cosmology is therefore simple and elegant, in the sense thatobservable quantities, such as the luminosity distance d L and the redshift dependenceof the Hubble constant H , take on analytic forms: d L = R h ( t )(1 + z ) ln(1 + z ) , (2)and H ( z ) = H (1 + z ) , (3)where z is the redshift, R h ( t ) = c/H ( t ), and H is the value of the Hubble constanttoday. Relations such as these have been tested using a broad range of measurements,such as the BAO observations described above, and have thus far accounted for thedata better than their counterparts in ΛCDM [28, 42, 43, 44, 45, 46, 47, 48, 49, 51,50, 52, 53, 54, 55, 56].Nonetheless, with its empirical approach, ΛCDM has done remarkably well as areasonable approximation to R h = ct in restricted redshift ranges. For example, usingthe ansatz ρ = ρ m + ρ r + ρ Λ to fit the data, one finds that the parameters in thestandard model must have quite specific values, such as Ω m ≡ ρ m /ρ c ∼ .
27, where ρ c is the critical density [57]. However, with these parameters, ΛCDM then requires R h ( t ) ≈ ct today, which is in fact the baseline constraint in the R h = ct model.One concludes from this that the optimized ΛCDM cosmology describes a universalexpansion equal to what it would have been with R h = ct all along. And many otherindicators support the view that using ΛCDM to fit the data therefore produces acosmology almost, but not entirely identical, to R h = ct , in spite of the fact that withits many free parameters, ΛCDM could have had an entirely diverse set of expansionhistories.
3. Cosmological Dynamics
The Friedmann-Robertson-Walker metric is conventionally written in terms of thecomoving coordinates ( t, r, θ, φ ), and takes the form ds = dt − a ( t ) (cid:20) dr − Kr + r (cid:0) dθ + sin θ dφ (cid:1)(cid:21) , (4)where K is the spatial curvature constant. In R h = ct , K must be zero in order for thegravitational radius to coincide with the Hubble radius, which the data (interpreted inthe context of ΛCDM) all seem to be confirming as well. We will therefore henceforthset K = 0 in all our derivations.The dynamical equations for this background FRW metric are obtained fromEinstein’s equations, G αβ ≡ R αβ − g αβ R = − πGT αβ , (5)where g αβ are the metric coefficients, and R αβ and R are the Ricci tensor and scalar,respectively: H ≡ (cid:18) ˙ aa (cid:19) = 8 πG ρ , (6)and ¨ aa = − πG ρ + 3 p ) , (7)where a dot denotes a derivative with respect to t , and ρ and p are, of course, theproper energy density and pressure in the co-moving frame. Throughout this paper,we work with natural units, in which ~ = c = 1. The continuity equation for the(perfect fluid) energy-momentum tensor, T αβ = ( ρ + p ) u α u β − pg αβ , (8)in terms of the four-velocity u α , yields a third (though not independent) equation,expressing the (local) conservation of energy:˙ ρ = − H ( ρ + p ) . (9) In the inflationary model, one assumes the existence of scalar inflaton fields thatdominate ρ in the cosmic fluid prior to the onset of leptogenesis and baryogenesis, withproperties that lead to an exponential solution for a ( t ) in Equations (6) and (7), thusheralding a very brief period of de Sitter expansion [58]. It is not difficult to imaginesuch fields influencing cosmological dynamics in the very early universe. For example,a grand unified theory based on the group SO(10) implies the existence of ∼ R h = ct universe does not need or have inflation, and therefore doesnot require the presence of an ‘inflaton’ field, we will nonetheless assume that atleast one scalar field dominated the cosmological dynamics at the very beginning.But unlike the situation with the standard model, the expansion factor a ( t ) inthe R h = ct universe must always be proportional to t , so the expansion in thiscosmology is never inflated. To clearly distinguish the hypothesized scalar field φ associated with the expansion in this model from those generally categorized as‘inflaton’ fields, we will therefore refer to it informally as the ‘numen’ field, giving riseto the earliest manifestation of substance in the nascent universe, with an equation-of-state ρ φ + 3 p φ = 0 and, as we shall see very shortly, whose quantum fluctuationsmight have seeded the subsequent formation of large-scale structure.A crucial difference between the R h = ct universe and other FRW cosmologiesis that the cosmic fluid in the former has strictly ‘zero active mass,’ meaning that ρ +3 p = 0, so the expansion proceeds without any net gravitational influence (after all,this is why ˙ a =constant). This suggests that coupling the numen field non-minimallyto gravity using the simple prescription ξ R φ / ξ is a dimensionlesscoupling constant) in the Lagrangian density may not be as relevant here as it is inthe inflaton case (though not impossible, of course). We will rely on the simplestassumption we can make, i.e., that the background dynamics is dominated by a singlehomogeneous minimally-coupled scalar field with action S = Z d x √− g L ( φ, ∂ µ φ ) , (10)where √− g = a ( t ) for the metric in Equation (4), and the Lagrangian density isgiven as L = m π R + 12 ∂ µ φ∂ µ φ − V ( φ ) , (11)where m P ≡ G − / is the Planck mass. As it turns out, the potential V ( φ ) for thenumen field φ is unique in R h = ct , and we shall derive it very shortly.Since the (background) field is homogeneous, we can ignore spatial gradients,and so the corresponding energy density ρ φ (= T ) and pressure p φ (= T ii ) are givensimply as ρ φ = 12 ˙ φ + V ( φ ) , (12)and p φ = 12 ˙ φ − V ( φ ) . (13)The zero active mass condition therefore immediately constrains the potential to havethe unique form V ( φ ) = ˙ φ , (14)and the energy conservation Equation (9) gives¨ φ + 3 H ˙ φ + ∂V∂φ = 0 , (15)the usual Klein-Gordon equation.The Friedmann Equation (6) similarly reduces to a very simple form, H = 4 πm V ( φ ) , (16)and combining this with Equation (14) then allows us to find an exact solution for thenumen field: φ ( t ) − φ ( t i ) = m P √ π ln (cid:18) tt i (cid:19) , (17)where t i is some fiducial time at which the field has an amplitude φ ( t i ). Returning toEquation (14), we therefore also have an exact—and unique—solution for the numenpotential: V ( φ ) = V exp ( − √ πm P φ ) (18)where, for convenience, we have defined the constant V ≡ m π t i exp ( √ πm P φ ( t i ) ) . (19)It is interesting to note that before settling on de Sitter (or quasi de Sitter)expansion for the inflationary paradigm, there were attempts in the 1980’s to considerminimally coupled inflaton fields with an exponential potential V p ( φ ) = V exp (cid:26) ± p √ πm P φ (cid:27) , (20)as a means of producing so-called power-law inflation (PLI) with p > a ( t ) ∝ t p . (21)Again, the intention with these was to circumvent the problems arising fromdeceleration in standard big bang cosmology. The numen-field potential (Equation 18)is clearly a special member of this class, though with p = 1 it does not inflate.Exponential potentials such as these are generally motivated in the context of Kaluza-Klein cosmologies [62], and arise in string theories, supergravity, and actually anytheory based on a conformal transformation to the Einstein frame.A difficulty commonly encountered with inflaton models is that they lack anexit mechanism for a decelerating (radiation or matter dominated) phase to succeedinflation. In contrast, the expansion rate is always constant in R h = ct , so nodynamical transition is required as the numen field decays into radiation and otherparticles in the standard model (via channels yet to be determined).
4. Cosmological Perturbations
We have learned from inflationary cosmology that to properly interpret anisotropiesin the CMB, one needs a description of the fluctuations characterized by severalobservables. These include: (1) the scalar spectral index n s , (2) the spectral index n T of the tensor perturbations and (where possible) (3) the tensor-to-scalar ratio r ,giving the ratio of tensor to scalar amplitudes [67, 68]. The Wilkinson MicrowaveAnisotropy Probe (WMAP) [21] and Planck [23] have placed strong bounds on atleast some of these parameters: n s = 0 . ± .
006 (corresponding to essentially ascale-free spectrum) and r < .
11 at 95% CL.Let us now consider small perturbations about the homogeneous numen field φ ( t ), φ ( t, ~x ) = φ ( t ) + δφ ( t, ~x ) , (22)keeping only terms to first order in δφ . The inhomogeneity implied by thesefluctuations requires that we also include metric perturbations about the spatiallyflat FRW background metric, which can be conveniently split into scalar, vector, andtensor components, depending on how they transform on spatial hypersurfaces.By now, it is well known that vector perturbations have no lasting influence forscalar fields. The perturbed FRW spacetime for the remaining linearized scalar andtensor fluctuations is therefore described by the line element [12, 69, 70, 71] ds = (1 + 2 A ) dt − a ( t )( ∂ i B ) dt dx i − a ( t ) [(1 − ψ ) δ ij + 2( ∂ i ∂ j E ) + h ij ] dx i dx j , (23)where indices i and j denote spatial coordinates, and A , B , ψ and E describe thescalar degree of metric perturbations, while h ij represent the tensor perturbations.This form follows the notation of Ref. [71], aside from the use of the symbol A insteadof φ inside the lapse function (since we are reserving this symbol to represent thescalar field in this paper).The Einstein equations for the scalar and tensor parts decouple to linear order, butthe form of the scalar equation is gauge dependent. However, one can identify a varietyof gauge-independent combinations of the scalar perturbations from within certaincoordinate systems. For example, in the comoving frame, the curvature perturbationΘ on hypersurfaces orthogonal to comoving worldlines may be defined [69] as a gaugeinvariant combination of the metric perturbation ψ and the scalar field perturbation δφ : Θ ≡ ψ + (cid:18) H ˙ φ (cid:19) δφ . (24) Expanding Θ in Fourier modes,Θ( t, ~x ) = Z d ~k (2 π ) / Θ k ( t ) e i~k · ~x , (25)where k is the comoving wavenumber, and using the linearized version of Einstein’sEquations (5) with the linearized metric in Equation (23), one arrives at the perturbedequation of motionΘ ′′ k + 2 (cid:18) z ′ z (cid:19) Θ ′ k + k Θ k = 0 , (26)where overprime now denotes a derivative with respect to conformal time dτ ≡ dt/a ( t ).In the R h = ct universe, we have a ( t ) = ( t/t ), where t is a fixed time usually takento be the present age of the Universe, so that a ( t ) = 1. Therefore τ ( t ) = τ i + t ln (cid:18) tt i (cid:19) , (27)where τ i is the conformal time at some fiducial cosmic time t i . To simplify the notation,we will define the zero of conformal time to be at t , so throughout this paper we willemploy the relation τ ( t ) = t ln a ( t ) . (28)The quantity z in Equation (26) is defined by the expression z ≡ a ( t )( ρ φ + p φ ) / H . (29)0Using Equations (6), (12) and (13) for the numen field, z reduces to the much simplerform z = m P √ π a ( t ) , (30)and so z ′ /z = 1 /t and z ′′ /z = 1 /t .The quantity z ′ /z typically depends on the background dynamics, so one oftenconveniently rewrites Equation (26) in terms of the so-called Mukhanov-Sasaki variable u k ≡ z Θ k [72, 73]. It is not really necessary to do this here since z ′ /z and z ′′ /z areactually constant for the numen field, but we will take this step anyway just to makeit easier to directly compare the differences between our solution and that pertainingto conventional inflaton fields. With this change of variable, the equation governingthe curvature perturbation now becomes u ′′ k + α k u k = 0 , (31)where α k ≡ t s(cid:18) πR h λ k (cid:19) − t s(cid:18) kR h a (cid:19) − , (32)and λ k ≡ πa/k is the proper wavelength corresponding to comoving wavenumber k .This expression for the ‘frequency’ α k is critical to understanding the nature ofquantum fluctuations in the numen field. Its most distinct departure from inflatonfields is that both the gravitational radius R h = t , and the proper wavelength λ k ∼ a ( t ), scale with time in exactly the same way, so the ratio R h /λ k or, equivalently, kR h /a , is constant. In this cosmology there is no crossing of wave modes back andforth across the horizon. In fact, once the wavelength of a mode is established whenit emerges into the semi-classical universe, it remains a fixed fraction of the Hubbleradius while both expand with time. And notice that all modes u k with a wavelengthsmaller than the horizon, R h > λ k / π , oscillate, while those with a super-horizonwavelength do not. (In this particular regard, the numen and inflaton fields behavesimilarly.) The analytic solution to Equation (31) may be written as follows: u k ( τ ) = (cid:26) B ( k ) e ± iα k τ (2 πR h > λ k ) B ( k ) e ±| α k | τ (2 πR h < λ k ) (33)The amplitude B ( k ) is fixed by an appropriate choice of vacuum, related to howthese modes are “born.” Quantum fluctuations of the inflaton field are created witha wavelength much smaller than the horizon, so they behave at first like an ordinaryharmonic oscillator (similar to the oscillatory solution in Equation 33). But duringinflation, H is essentially constant, so λ k overtakes the Hubble radius R h = 1 /H and becomes much larger, and the mode becomes an overdamped oscillator, with anamplitude that approaches a constant value, a process often referred to as “freezing.”Once inflation has ended, the Hubble radius resumes its rate of growth and overtakes λ k , which is said to then “re-enter” the horizon.In a time-independent spacetime, a preferred set of mode functions, and thereforean unambiguous physical vacuum, may be defined by minimizing the expectation valueof the Hamiltonian. In Minkowski space, this means taking the positive frequencymode u k ∼ e − ickτ , i.e., the minimal excitation state, and setting B ( k ) = 1 / √ k [12]. This prescription, however, does not usually generalize straightforwardly to time-dependent spacetimes, but this vacuum ambiguity can still be resolved in inflationarymodels by arguing that in the remote past all observable modes had a wavelength1much smaller than the horizon, and were therefore not affected by gravity, so theirfrequencies were essentially time-independent. They therefore behaved as they wouldin Minkowski space. This approach defines the preferred set of mode functions anda unique physical vacuum known as the Bunch-Davies vacuum [74]. The amplitude B ( k ) for super-horizon modes is then evaluated from the Bunch-Davies normalizationby equating the amplitudes before and after freezing. And because the value of a ( t )at which the freezing occurs is proportional to k (via the condition k/a ∼ /R h ),this process results in a scale-free spectrum (see below), consistent with the observedanisotropies in the CMB, considered to be one of the strongest factors in favor of theinflationary paradigm.However, some have questioned the fundamental basis for this picture becausein many models of inflation, the de Sitter phase lasted so long that the inflatonmodes responsible for the creation of large-scale structure would have been born withwavelengths much shorter than the Planck scale (and therefore well before the Plancktime), where the use of semi-classical physics is uncertain. This “trans-Planckianissue” [75] revolves around the question of whether the semi-classical description ofour Universe breaks down prior to the Planck time, set by the condition that theCompton wavelength λ C ≡ π/m of a mass m be equal to its Schwarzschild radius R h ≡ Gm . Current physics may have to be modified on spatial scales smaller thanthe resulting Planck length λ P ≡ λ C ( m ) at the specific value of m where this equalityis reached: λ P = √ πG . (34)The corresponding Planck time t P is simply this Planck length divided by the speedof light c . Numerically, we have λ P ≈ . × − cm and t P ≈ . × − s. Thesedefinitions actually make more sense for the R h = ct universe than they do for thestandard inflationary model, because the gravitational radius R h is in fact equal to t ,so the Planck time is simply the age of the Universe when the Hubble radius equaledthe Planck scale (i.e., R h = λ P ).Let us now track the mode growth associated with the CMB anisotropies inthe R h = ct universe back to these earlier times and see how they are related to λ P and t P in this cosmology. The CMB spectrum has features ranging from sub-degree scales to tens of degrees. The Sachs-Wolfe effect [76], responsible for couplingthe metric fluctuations with the primordial perturbations, contributes to temperatureanisotropies on all scales, but tends to dominate at angles & ◦ − ◦ . On sub-degree scales, the spectral peaks are primarily dependent on the pressure and densityvariations associated with baryon acoustic oscillations. The characteristic CMB scalerepresenting the effects of scalar/metric fluctuations therefore appears to be ∼ ◦ − ◦ .In the R h = ct cosmology, the angular-diameter distance is given as [25, 26, 50] d A = R h ( t )(1 + z ) ln(1 + z ) . (35)Therefore, a θ -fluctuation at redshift z CMB corresponds to a proper wavelength λ θ ( z CMB ) = 2 π (cid:18) θ ◦ (cid:19) R h ( t )(1 + z CMB ) ln(1 + z CMB ) . (36)And with a ( t ) = t/t , it is straightforward to see that at the Planck redshift, z P ≡ t /t P −
1, the numen-field mode responsible for this anisotropy had a correspondingwavelength λ θ ( z P ) given by the expression λ θ ( z P ) λ P ≈ (cid:18) θ ◦ (cid:19) ln(1 + z CMB ) . (37)2A precise estimate for z CMB does not exist yet for the R h = ct universe, but thisuncertainty has negligible impact on the use of Equation (37) because the behaviorof d A with redshift in this cosmology renders λ θ ( z P ) /λ P only weakly dependent onthe redshift at last scattering. This may be seen in Table 1, where we quote theratio λ θ ( z P ) /λ P for two angles ( θ = 1 ◦ and 10 ◦ ) and a very broad range of CMBredshifts. Clearly, the numen-field fluctuations producing the CMB anisotropies hada size comparable to the Planck scale were we to trace them back to the Plancktime. The significance of this feature should not be underestimated. In the R h = ct cosmology, the Universe underwent an expansion by over 60 orders of magnitude in a ( t )between z P and z CMB . Yet in this model the observed scale characterizing the CMBanisotropies tracks back directly to the Planck length at t P , in contrast to standardinflationary cosmology in which the CMB fluctuations have no obvious connection tothe Planck scale. It would be a remarkable coincidence for λ ◦ ( z P ) ∼ λ P if these twoscales were not related dynamically in some way. Table 1.
Perturbation wavelength at t P producing a ∼ ◦ -10 ◦ fluctuation in the CMB z CMB λ ◦ ( z P ) /λ P λ ◦ ( z P ) /λ P
500 0 .
11 1 . .
12 1 . .
16 1 . R h = ct , it is therefore quite natural—perhaps even required—forus to view the modes as having emerged into the semi-classical Universe starting at thePlanck scale λ P . Such an idea—that modes may have been born at a specific physicalscale—has already been considered by several other authors, particularly Hollands andWald [77], who focused on the question of where and when a semi-classical descriptionof our Universe may be valid. Their context was different from ours, and it was notclear why the physical scale they introduced (which they called l ) ought to somehowbe related to λ P . They found that to match the observed fluctuation amplitude in theCMB, they needed l to be five orders of magnitude larger than the Planck length. Asthey noted, however, and as we shall see below, the Hollands-Wald concept works ina way that makes this ratio essentially independent of the behavior of a ( t ), so we willalso conclude that although the numen-field fluctuations might have begun across thePlanck region, their emergence into the semi-classical universe could not have beencompleted on a scale length shorter than λ ∼ λ P .The Hollands-Wald concept for how quantum fluctuations are born in this contextis based on the assumption that semi-classical physics applies (at least in some roughsense) to phenomena on spatial scales larger than this fundamental length λ , sothat modes effectively emerge only when their proper wavelength equals λ . (Note,however, that the idea of modes being created when their wavelength is at a givenspatial scale is actually not unique. Some previous arguments supporting this conceptmay be found in Refs. [78, 79].) In this view, it makes sense to talk about a classicalspacetime metric and quantum fields at times earlier than the Planck time t P , butonly if this is done with a restriction to phenomena based solely on spatial scales largerthan λ . In this picture, k -modes may be created at different times rather than all at3 L e n g t h Cosmic time tt
P k . . R = ct h λ = k a(t)k λ λ P Figure 1.
Schematic diagram illustrating a k -mode born across the Planck regionand emerging into the semi-classical universe when its wavelength λ k equals thescale λ . The vertical axis shows proper distances as a function of cosmic time t (increasing to the right on the horizontal axis). The time of birth t k is definedby the condition λ = 2 πa ( t k ) /k . In the R h = ct cosmology, the ratio λ k /R h isconstant for all time t . once, though in a sequence based on the relationship between λ k and λ .In the R h = ct cosmology, we have several good reasons for adopting Hollandsand Wald’s central idea. Chief among them is the empirical evidence described above,which supports the conclusion that 10 ◦ -anisotropies in the CMB would have had asize ∼ λ P at ∼ t P (see Table 1). Second, the very notion of a Planck length rests onthe physical limitations imposed on the localization of a defined mass by its Comptonwavelength λ C . Since the gravitational radius R h ( t ) defines the maximum size ofany causally connected region at time t in this model [82, 83, 84], only proper massessmaller than ∼ R h / G have any physical meaning. So quantum mechanical “fuzziness”extends over a scale ∼ λ C ( t ) = λ /t , even bigger than λ P at t < t P .As is well known from our experience with fluctuations in the inflaton field, andas evident in the form of the frequency α k in Equation (32), modes with λ k < πR h oscillate, while those with λ k > πR h are effectively “frozen” on super-horizon scales(see discussion in § t cmb . So we follow Hollands & Wald (2002) inidentifying the Sachs-Wolfe perturbations in the CMB with those trans-Planckianfluctuations with super-horizon wavelengths at the time they were born. Figure 1shows the key scales relevant to this hypothesis, including the Planck wavelength λ P = R h ( t P ), and the wavelength λ k = 2 πa ( t ) /k of mode k . We emphasize againthe key difference between the numen-field and inflaton fluctuations, in that the ratio λ k /R h is constant for the former, while it first increases and then decreases duringinflation.4For these reasons, we adopt the view that all the k -modes of interest in the CMBsatisfy the condition k = 2 πǫ k λ P a ( t P ) , (38)where ǫ k &
1. They emerge into the semi-classical universe when their wavelengthequals the scale λ , so that (with a ( t ) = t/t ) t k = k λ t π . (39)An alternative way to write this is t k = λ ǫ k λ P t P . (40)We now write Equation (32) using the approximate expansion α k ≈ k (cid:18) − kt ) (cid:19) , (41)so that with the definition of z in Equations (29) and (30), the metric perturbationΘ k has an amplitude frozen at the birth time t k , given by the expression (cf. ref. [77]) | Θ k | = (cid:18) H ˙ φ (cid:19) a ( t ) α k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t k ≈ π k λ λ (cid:18) − kt ) (cid:19) − . (42)The power spectrum for these curvature perturbations is given by the k -spaceweighted contribution of modes [3, 12, 80, 81], commonly written as P Θ ( k ) ≡ k π | Θ k | . (43)We can therefore confirm that the spectrum of numen scalar curvature perturbations isalmost scale free, i.e., P Θ ( k ) ∼ k , as seen in the CMB fluctuations, but not exactly.The common way to quantify the deviation from scale-invariance is via the scalarspectral index, n s , defined according to n s − ≡ d ln P Θ ( k ) d ln k . (44)From Equations (42-44), we find that n s ∼ − kt ) − . (45)As we noted earlier in this section, the observed index is n s = 0 . ± .
006 [23],suggesting that the actual power spectrum is only approximately scale free. Theimplication of this measurement for the numen-field fluctuations is that the slightdeviation from a pure scale-free spectrum appears to be due to the difference k − α k in Equation (32). Furthermore, as was the case for Hollands and Wald [77], we findthat the correct amplitude of the fluctuations in the CMB is produced if we choose λ to be of order 10 λ P , the grand unification scale.Of course, the value of the ratio in Equation (45)—and therefore of the inferredscalar spectral index n s —depends on the wavenumber k . So the numen fluctuationspectrum will have a weakly running spectral index. Though perhaps not as reliable5as the index n s itself, Planck constrained its scale dependence dn s /d ln k to have thevalue − . ± . dn s d ln k ∼ kt ) [2( kt ) − , (46)a very small, though positive number. Future work will tell whether this differenceis meaningful, or whether it is simply due to the fact that the analysis reported by Planck was carried out solely in the context of ΛCDM.
The tensor perturbations h ij are transverse ( ∂ i h ij = 0) and trace-free ( δ ij h ij = 0) andare automatically independent of coordinate gauge transformations. These representgravitational waves evolving independently of linear matter perturbations, and aretypically decomposed into eigenmodes e ij of the spatial Laplacian operator withcomoving wavenumber k and scalar amplitude h ( t ), such that h ij ( t, ~x ) = h ( t ) e (+ , × ) ij ( ~x ) , (47)with two independent polarization states + and × .In addition to decoupling completely from scalar perturbations and not providingany backreaction to the metric, gravity waves also satisfy sourceless equations whenthe energy-momentum tensor is diagonal, like that in Equation (8). With the definition v k ( t ) ≡ ( m P / √ π ) ah k ( t ) for the Fourier components h k ( t ) of h (based on Equation 47and our definition of the Planck mass m P = G − / ), it is easy to see that the modeequation for the tensor perturbations (analogous to Equation 31) is v ′′ k + α k v k = 0 , (48)with the same frequency α k defined in Equation (32).The fields v k and u k are considered to have similar attributes, notably, that bothare canonically normalized, and that both ‘freeze’ when their wavelengths exceed thehorizon scale. Therefore, we can immediately write down the expression equivalent toEquation (42) for the amplitude of h k : | h k | = 32 πm a α k (cid:12)(cid:12)(cid:12)(cid:12) λ T , (49)where λ T is the scale—analogous to λ for the scalar perturbations—at which thetensor modes emerge into the semi-classical universe. And therefore since there aretwo independent tensor polarization states, the tensor power spectrum (analogous toEquation 43) is P T ( k ) ≡ k π | h k | = 16 λ λ (cid:18) − kt ) (cid:19) − . (50)We are now in a position to examine the third observational signature typicallyassociated with the idea of a quantum-fluctuation origin for the anisotropies in theCMB—the ratio of tensor to scalar power, which we may write as follows: r ≡ P T ( k ) P Θ ( k ) = 16 (cid:18) λ λ T (cid:19) . (51)6Those familiar with the inflationary scenario will recognize that this expression is verysimilar to the result associated with an inflaton field, except that in that case the right-hand side of this expression is 16 ǫ , in terms of the slow-roll parameter ǫ ≡ − ˙ H/H .Because tensor modes would have decoupled completely from everything else,one does not know about these fluctuations (1) whether they were produced in thetrans-Planckian region, (2) whether they emerged into the semi-classical universe atthe same fixed length scale λ as the scalar perturbations, or (3) whether they wereeven generated after the Planck time. If they did emerge at a fixed scale λ T , theywould likely have a near-scale free spectrum, like the curvature perturbations, butone could not predict their power relative to that of the scalar fluctuations withoutknowing something about the scale at which they emerged [77]. It may already bepossible to eliminate the first possibility, since the scaling for Θ k and h k , through theirdefinition in terms of the modes u k and v k , would suggest a ratio of tensor to scalarpower exceeding current upper limits. Indeed, if we adopt the value r . .
11 as themost recent observational constraint, our model would require λ T & λ , (52)meaning that any gravity waves generated in this picture would also have beenproduced at the GUT scale. In the case of inflation, the slow-roll parameter ǫ isa direct probe into the energy scale of the inflaton field, and it is generally understoodthat if r & .
01, then the inflaton potential has a value V / ∼ ( r/ . / GeV,which itself lies in the GUT energy range. In some ways, this convergence of ideas israther promising for the quantum-perturbation model for the origin of fluctuations inthe cosmic fluid, since it suggests that the physics (as we know it) of scalar fields inthe early Universe is rather tightly constrained, and perhaps the simplest extensionsto the standard model are on the horizon.
5. Conclusions
One of the strongest arguments in favor of the freeze-out mechanism during inflationis the coherence of the observed CMB fluctuations [85]. Curvature perturbationseventually source density fluctuations that evolve under the influence of gravity andpressure to produce the CMB inhomogeneities and subsequent large-scale structure.If one reasonably supposes that recombination happens instantaneously (at least incomparison to the evolutionary timescale), then fluctuations with different wavelengthsinfluence the surface of last scattering at different phases in their oscillations. However,if all Fourier modes of a given wavenumber have the same phase, then they interferecoherently, resulting in a CMB spectrum with clearly defined peaks and troughs.Without this coherence, the various modes would all combine to produce white noise.With inflation, all the mode phases are set when the fluctuations exit the horizon,which therefore remain coherent upon subsequent re-entry. Something very similar tothis happens with the numen field, since all the scalar modes of a given wavenumber k emerge into the semi-classical universe at the same scale λ and, therefore, at the sametime t k . These modes are super-horizon and frozen during the subsequent expansion.They eventually source matter and radiation fluctuations with the same phase whenthe numen field decays into standard-model particles. So the mechanism for generatingcoherence of the numen modes is very similar to that of the inflaton field, thoughperhaps a little simpler since it requires fewer steps and is a natural extension ofthe wavenumber-dependence of the emergence of these modes, unlike those associated7with the inflaton field, which are considered to have been born at arbitrarily early(pre-Planckian) times [78].The suggestion is sometimes made that Planck-era physics may eventually bestudied with the CMB. In the R h = ct universe, this idea is more than merespeculation. Indeed, as we have shown in this paper, CMB fluctuation scalesand amplitudes are preserved at the values they had as they emerged out of thePlanck domain. In particular, the identification of the angular scale of the CMBinhomogeneities with the Planck length is a strong factor in favor of this cosmology,particularly since the Universe would have expanded by over 60 orders of magnitudebetween the Planck and recombination times. One cannot completely rule out acoincidence such as this, though the probability of its occurrence is extremely small.So the connection between the CMB and Planck scales is already clearly defined in R h = ct .We have also seen that if matter in the early universe was dominated by asingle scalar field, then its potential in this model is known precisely (and givenin Equation 18). This result may motivate further exploration of Kaluza-Kleincosmologies, string theories, and supergravity, in which exponential potentials suchas these are well justified. In concert with constraints imposed by CMB observations,particularly the value of the scalar spectral index n s , there is therefore hope that newphysics may emerge with relevance to the trans-Planckian domain. Acknowledgments
It is a pleasure to acknowledge helpful discussions with Bob Wald, Sean Fleming,Robert Caldwell, Daniel Sudarsky, and Robert Brandenberger. Some of this workwas carried out at Purple Mountain Observatory in Nanjing, China, and was partiallysupported by grant 2012T1J0011 from The Chinese Academy of Sciences VisitingProfessorships for Senior International Scientists.
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