aa r X i v : . [ h e p - t h ] J un Inflation, quintessence, and the origin of mass
C. Wetterich
Institut f¨ur Theoretische PhysikUniversit¨at HeidelbergPhilosophenweg 16, D-69120 Heidelberg
In a unified picture both inflation and present dynamical dark energy arise from the same scalarfield. The history of the Universe describes a crossover from a scale invariant “past fixed point”where all particles are massless, to a “future fixed point” for which spontaneous breaking of theexact scale symmetry generates the particle masses. The cosmological solution can be extrapolatedto the infinite past in physical time - the universe has no beginning. This is seen most easily in aframe where particle masses and the Planck mass are field-dependent and increase with time. Inthis “freeze frame” the Universe shrinks and heats up during radiation and matter domination. Inthe equivalent, but singular Einstein frame cosmic history finds the familiar big bang description.The vicinity of the past fixed point corresponds to inflation. It ends at a first stage of the crossover.A simple model with no more free parameters than ΛCDM predicts for the primordial fluctuationsa relation between the tensor amplitude r and the spectral index n, r = 8 . − n ) − . I. Introduction
A scalar field plays a dominant role both for inflationin primordial cosmology and dynamical dark energy in thepresent epoch. The potential of this field constitutes pri-mordial or late dark energy, driving an accelerated expan-sion in the big bang picture. Quintessential inflation [1, 2]identifies the inflaton field for inflation with the scalar fieldof quintessence or cosmon which is responsible for presentdynamical dark energy. In particular, cosmon inflation [3]formulates this unification in the context of variable grav-ity [4], where the strength of gravity depends on the valueof the cosmon field.Both inflation and quintessence can be closely relatedto approximate dilatation or scale symmetry. For inflationthis symmetry is at the origin of the observed approximatescale invariance of the spectrum of primordial fluctuations.For present dynamical dark energy the cosmon plays therole of the pseudo Goldstone boson of spontaneously bro-ken dilatation symmetry [5]. In case of exact dilatationsymmetry it would be an exactly massless dilaton, while atiny mass and potential are generated by a “scale symmetryviolation” or “dilatation anomaly”. Scale symmetry is in-timately related to fixed points of “running” dimensionlesscouplings or mass ratios. At a fixed point any informationabout intrinsic mass or length scales is lost. Quantum scalesymmetry is then realized even if the underlying quantumfield theory is not scale invariant.The presence of approximate scale symmetry both forthe primordial and late cosmology suggests that the infi-nite past and infinite future of the universe correspond tofixed points. We propose here that the two fixed pointshave different properties. For the fixed point in the infi-nite past scale symmetry is not spontaneously broken. Allmasses vanish. In contrast, the fixed point that will be ap- proached in the infinite future is characterized by sponta-neous symmetry breaking of dilatation symmetry, resultingin a spectrum of massive particles and a massless dilaton.The way how scale symmetry is realized and explicitly orspontaneously broken is directly related to the basic originof mass. All particle masses are generated either by explicitor spontaneous breaking of scale symmetry. The explicitbreaking by an intrinsic mass scale plays a crucial role forthe crossover between the two fixed points. It is responsiblefor scale violation in the primordial fluctuation spectrumand for the end of inflation. Spontaneous breaking charac-terizes the “future fixed point” and our present universe.The presently observed particle masses are dominated byspontaneous scale symmetry breaking, while dark energyreflects the tiny explicit breaking. The basic mechanismsthat generate the particle masses thus provide the physical“raison d’ˆetre” for inflation and late dark energy, such thatthese key cosmological ingredients appear less “ad hoc”.This work is motivated by a central assumption aboutthe properties of quantum gravity that we call “crossovergravity”. The running of dimensionless couplings or massratios as a function of some intrinsic mass scale µ is as-sumed to exhibit two fixed points for µ → ∞ and µ → µ . Dimensionless functions can only depend on dimension-less quantities. If time and space gradients or momenta areproportional to µ (or can be neglected), the renormalizeddimensionless functions can still depend on the ratio µ/χ ,with χ the value of the scalar cosmon field which equalsthe variable Planck mass in our normalization. The ul-traviolet (UV) field point is realized for µ → ∞ at fixed χ or χ → µ . Indeed, with all particle massesproportional to χ this fixed point realizes unbroken scalesymmetry. All excitations are massless. The infrared (IR)fixed point occurs for µ → chi → ∞ . A nonvanishingvalue of χ spontaneously breaks scale symmetry. We willsee that the cosmological solutions of our model realize anevolution where χ vanishes in the infinite past and divergesin the infinite future. The cosmological evolution thereforeinterpolates between the UV-fixed point in the past andthe IR-fixed point in the future.Inflation describes the vicinity of the past fixed point.It can extend to the infinite past in physical time. Theinflationary epoch has to end, however. ”Late cosmology”comprises epochs of radiation-, matter- and dark energydomination. It is characterized by the approach towardsthe future fixed point. We will describe the transition frominflation to late cosmology as a first stage of the crossoverbetween the two fixed points. In the crossover region cou-plings have to run from one fixed point to the other. Scalesymmetry is therefore necessarily violated in the crossoverregion. This is the basic reason for the qualitative changein the dynamics of the cosmon that occurs at the end ofinflation.If there is more than one relevant or marginal devia-tion from the “past fixed point” the crossover may occurin different stages. In case of a slow running (e.g. loga-rithmic) the scales associated to these stages can be sepa-rated by many orders of magnitude. We assume here thatin the beyond standard model sector of particle physicsthe crossover is completed only in a second stage. Thissector influences the masses of the neutrinos by “non-renormalizable operators” according to the see-saw or cas-cade mechanism. While the mass ratios of all particlesexcept for neutrinos reach fixed values already at the endof inflation, the ratio of neutrino mass to electron massmakes the transition to the future constant value only inthe present epoch. The relative increase of the neutrinomasses realizes “growing neutrino quintessence” [6, 7] andexplains the “why now problem” by relating the presentdark energy density to the present neutrino mass.The history of dark energy reflects the two stages ofthe crossover. A primordial scaling solution correspondsto dominant dark energy during inflation. The first stageof crossover ends this scaling solution, triggering a tran-sition to a different scaling solution during the radiationand matter dominated epochs. As a consequence of thisscaling solution dark energy decreases proportional to thedominant radiation or matter component [5], constitutinga small fraction of “early dark energy”. Neutrinos are rel-ativistic during this epoch and their masses play no role.The second stage of the crossover takes place in the presentcosmological epoch. A substantial increase of the neu-trino masses ends the second scaling solution once neutri-nos become non-relativistic. This cosmic “trigger event”has happened around redshift z ≈
5, inducing a transitionepoch with dominant dark energy and accelerated expan-sion. Once the second stage of the crossover is completed,the ratio between neutrino and electron mass approachesa constant value according to the future fixed point. Cos-mology in the far future may correspond to a new scalingsolution for which dark energy needs not to remain domi-nant.A crossover in two steps can be associated with a flow trajectory in the vicinity of an intermediate (approximate)fixed point. We may refer to this fixed point as the “stan-dard model fixed point”. For this fixed point the renormal-izable dimensionless couplings of all particles are the onesobserved in present experiments. Neutrino masses, how-ever, are typically substantially smaller than their presentvalue. The standard model fixed point may be unstable inthe sector of heavy particles with masses much larger thanthe Fermi scale, or in a sector of standard model singletscoupled only very weakly to the particles of the standardmodel. Such an instability will finally drive the flow trajec-tory away from the standard model (SM) fixed point andtowards the infrared fixed point.On the other hand, the zeros of the β -functions for therenormalizable couplings of the standard model are stablefor decreasing µ , such that the presently measured valueshold to high accuracy for the entire matter and radiationdominated epochs. The second step of the crossover af-fects first only the neutrino masses. Nevertheless, whenthe second step of the crossover will be completed in thefar future, it is possible that the changes in the beyondstandard model sector also affect the renormalizable cou-plings of the standard model. Their values at the infraredfixed point could be different from the present ones.We have depicted the flow trajectory in some abstract“coupling space” or “theory space” in Fig. 1. It showsthe first stage of the crossover from the UV-fixed pointto the vicinity of the standard model fixed point, and thesubsequent second step of the crossover to the infrared fixedpoint. We can associate the different cosmological epochsto the corresponding parts of the flow trajectory. SM Q CR2CR1 EI M IRRIUV X FIG. 1: Schematic view of the crossover from the UV-fixed point(UV) to the infrared fixed point (IR). Arrows indicate the di-rection of decreasing µ or increasing χ . This direction corre-sponds to the flow of cosmic time from the infinite past (UV) tothe infinite future (IR). The crossover trajectory passes near an(approximate) fixed point (SM) that characterizes the presentstandard model. The two regimes of fast changes, CR1 andCR2, correspond to the two steps of the crossover. We also in-dicate the corresponding cosmological epochs: inflation (I), endof inflation (EI), radiation domination (R), matter domination(M), dark energy domination (Q). This paper is organized as follows: In sect. II we in-troduce the flow equations underlying our approach. Theydescribe the change of couplings as an intrinsic overall massscale µ is varied. We discuss the properties of the ultravio-let and infrared fixed points. For this purpose we choose aframe of variable gravity where the crossover is describedby the flow equation for the “kinetial”, e.g. the coefficientof the scalar kinetic term. In particular, we investigatesettings where the kinetial diverges at the ultraviolet fixedpoint with a large anomalous dimension.In sect. III we turn to the cosmological solution forvalues of the cosmon field χ close to the ultraviolet fixedpoint and the first step of crossover away from it. It de-scribes an epoch of inflation and its end. We compute theproperties of the primordial density fluctuations. Both thespectral index n and the tensor to scalar ratio r are deter-mined by the anomalous dimension σ and therefore related,1 − n = r (2 + σ ) /
16. Computing r and n in terms of σ andthe number of e -foldings N between horizon crossing of theobservable fluctuations and the end of inflation, we estab-lish the relation r = 8 . − n ) − . µ over the crossoverscale m , which is exponentially small due to the slow run-ning near the fixed point.Sect. IV discusses “late cosmology” after the end of in-flation. It starts with a scaling solution for the radiationand matter dominated epochs that is characterized by asmall almost constant fraction of early dark energy [5, 8–10]. This scaling explains why the present dark energydensity is of the same order as the present matter energydensity. In particular, we discuss models where the infraredfixed point corresponds to a “conformal kinetic term”. Thedeviation from the fixed point is characterized by a function B ( χ/µ ) that decreases with an inverse logarithm for large χ/µ, B − = κ ln( χ/µ ). The fraction in early dark energyis proportional to B and therefore naturally small for thelarge values of χ relevant for late cosmology. The slow flowof B induces small scaling violations for the cosmologicalsolution that we discuss in terms of an approximate ana-lytic solution. We find a low value for the dark energy frac-tion at last scattering, close to the observational bounds.As the second step of the crossover sets in the neutrinomasses start to increase substantially. Once neutrinos be-come non-relativistic they stop the scaling solution, “freez-ing” the dark energy density at the value it has reached atthis moment. This leads to a phenomenology very close toa cosmological constant, with a value determined by thepresent average neutrino mass. Such a scenario solves the“why now?” problem.In sect. V we turn more closely to the particle physicsaspects of the ultraviolet fixed point. For an anomalousdimension in the range σ > µ -dependence of couplings in the quan-tum effective action to the scaling solutions for the effec-tive average action. We discuss the appearance of relevantparameters at the UV-fixed point as free integration con-stants in the scaling solution. In appendix B we illustratethe crossover between two fixed points in the flow of di-mensionless couplings or mass ratios. We discuss the timevariation of couplings in the standard model and neutrinomasses.Appendix C contains the field equations derived fromthe quantum effective action of crossover gravity. We in-clude higher order curvature invariants for the discussionof asymptotic solutions extending to the infinite past. Nu-merical solutions show the approach of a large class of so-lutions towards a family of scaling solutions. Some of thesolutions correspond in the Einstein frame to a transitionfrom pre-big-bang to big-bang cosmology, while they arecompletely smooth in the freeze frame. Appendix D re-formulates the model with a curvature squared invariantin terms of an explicit additional scalar field. This helpsto understand the properties of the solutions discussed inappendix C. Appendix E enlarges the class of crossovermodels and maps them to the freeze frame. II. Fixed points and crossover
In this section we display our model of “crossover grav-ity”. We discuss the ultraviolet and infrared fixed point fora system of gravity coupled to a scalar field. Away from thefixed point the dimensionless couplings are scale-dependentand realize a crossover between the two fixed points. Ourmain tool is the quantum effective action Γ from whichthe exact field equations follow by variation. The fixedpoints and the crossover are reflected in the properties ofΓ. Besides the masses and couplings of other particles ourmodel involves only four parameters which describe cos-mology from inflation to present dark energy domination.
1. Running couplings and fixed points
In quantum field theories the renormalized dimension-less couplings “run” as functions of an intrinsic mass scale µ . Here we consider all intrinsic mass parameters as be-ing proportional to µ , with ratios of intrinsic mass scalesassociated to dimensionless couplings. For a fixed pointthis flow stops and dimensionless couplings become inde-pendent of µ . An ultraviolet (UV) fixed point is reachedif suitable dimensionless couplings reach constant valuesfor µ → ∞ . Such a fixed point renders gravity non-perturbatively renormalizable (asymptotic safety [11–14]).Dilatation symmetry is an exact quantum symmetry at theUV fixed point. An infrared (IR) fixed point corresponds tothe stop of the flow of dimensionless couplings for µ → µ scale symmetryis again realized. In general, the existence of an IR fixedpoint is not compulsory - alternatives are diverging dimen-sionless couplings for µ → µ c >
0. We assume here that suchdivergencies do not happen and an IR-fixed point thereforeexists. A first functional renormalization investigation ofsuch a possible IR fixed point can be found in ref. [14].The flow of couplings as a function of µ is similar butnot identical to the running as a function of ˜ µ ∼ momen-tum divided by particle mass (say the electron mass). Itis this running as a function of momentum/mass that isdescribed by the usual β -functions of the standard modelof particle physics. There a non-trivial running typicallyoccurs in the range where ˜ µ is larger than the relevant par-ticle masses, while it stops once ˜ µ is below those masses. Inour setting, the flow as a function of µ describes the effectof a simultaneous change of all intrinsic mass scales ∼ µ .Besides the change in momentum scale this also includesthe change due to a mass parameter in the effective cosmonpotential. The µ -flow equations need a separate computa-tion which has not yet been performed. They partly aresimilar in spirit to the running of couplings as a functionof a mass parameter investigated by Symanzik [15]. Dif-ferent µ correspond conceptually to a family of differenttheories. These theories cannot be distinguished by obser-vation, however. Since only dimensionless ratios can beobserved the value of µ just sets the unit for quantitieswith dimension of mass or inverse length or time. (We use ~ = c = k B = 1). We will employ here µ − = 10 yr , (1)such that the present value of the variable Planck masstakes its usual value, cf. sect. IV. There is also some anal-ogy to the functional renormalization flow of the effectiveaverage action [16, 17], with IR-cutoff k associated to µ .We discuss the conceptual setting of the µ -flow equationin more detail in the appendix A and give examples inappendix B.
2. Variable gravity
We will work within variable gravity [4] and investigatethe cosmological solutions of the field equations derived from the quantum effective action for the coupled cosmon-gravity systemΓ = Z x √ g (cid:26) − χ R + µ χ + 12 (cid:0) B ( χ/µ ) − (cid:1) ∂ µ χ∂ µ χ (cid:27) . (2)The variable Planck mass is given by the value of the cos-mon field χ . The quadratic cosmon potential V = µ χ involves the intrinsic mass scale µ . A large family of ef-fective actions can be brought by field transformations toa form where the coefficient of the curvature scalar R is − χ and the scalar potential is quadratic, V ( χ ) = µ χ .We will discuss this issue in sect. VI. We then remain withthe dimensionless function B ( χ/µ ). Its dependence on µ isdescribed by the µ -flow equation. Stability requires B ≥ B = 0 , µ = 0.The quantum effective action should be supplementedby higher order curvature invariants,∆Γ = Z x √ g (cid:26) − C ( χ/µ ) R + D ( χ/µ ) (cid:18) R µν R µν − R (cid:19) (3)+ E ( χ/µ )( R µνρσ R µνρσ − R µν R µν + R ) (cid:9) . These terms will play a role for graviton-graviton scatter-ing at and near the UV-fixed point and for the approachof the cosmological solution to the infinite past, χ/µ → E the last term ineq. (3) is the topological Gauss-Bonnet invariant whichdoes not contribute to the field equations. The invariantmultiplying D can be written as a linear combination ofthe squared Weyl tensor and the Gauss-Bonnet invariant.For constant D the term ∼ D does not contribute to thefield equations for a spatially flat Robertson-Walker met-ric. The influence of the higher curvature invariant withconstant C is discussed in the appendices C and D.We do not include a possible scale invariant contributionto the cosmon potential ∆ V = λχ . Indeed, the functionalrenormalization investigation [14] of the behavior of a pos-sible fixed point suggests that the cosmon potential cannotincrease ∼ χ for χ → ∞ . The infrared fixed point pro-posed in ref. [14] has indeed λ = 0. A term ∼ λχ is scaleinvariant but not conformal invariant. If the scale invari-ance of the IR-fixed point for χ → ∞ implies conformalsymmetry, as advocated in ref. [18, 19], such a term is notallowed. This situation is suggested by the investigation ofhigher dimensional theories with scale invariance [20, 21].It is precisely the behavior V ( χ → ∞ ) ∼ χ that wouldnot allow a field redefinition to the freeze frame V = µ χ .(Other powers are allowed.) The absence of a term ∼ λχ implies the asymptotic vanishing of the observable cosmo-logical constant [5].Dimensionless functions as B (or C , D and E ) can onlydepend on the dimensionless ratio χ/µ . This links their µ -dependence according to the flow equation to their de-pendence on χ . The UV-fixed point for µ/χ → ∞ can alsobe seen as the limit χ →
0, while the IR-fixed point cor-responds to the limit χ → ∞ . We will find cosmologicalsolutions where χ varies from χ → χ → ∞ in the infinite future. This is how cosmology candescribe the crossover between two fixed points. The UV-fixed point for χ → χ → ∞ is associatedwith the “future fixed point”.In the context of variable gravity the flow equations fordimensionless couplings concern the quantities B, C , D and E . More on the conceptual status of the flow equations forthese quantities can be found in appendix A. In the presentpaper we assume a specific form of B ( χ/µ ) and investigatethe cosmological consequences of such a setting.
3. Infrared and ultraviolet fixed points
For the IR-fixed point µ vanishes and B reaches a con-stant, lim µ → B ( χ/µ ) = B ∞ . The term ∼ µ χ in eq. (2) isabsent for µ →
0, and the quantum effective action con-tains no longer any parameter with dimension mass. It isinvariant under scale symmetry, with a scaling of χ accord-ing to its canonical dimension. At the IR-fixed point theeffective action is scale invariant and takes the simple formof a free scalar field coupled to gravity,Γ IR = Z x √ g (cid:26) − χ R + 12 ( B ∞ − ∂ µ χ∂ µ χ (cid:27) . (4)(For χ → ∞ we can neglect the higher order curvatureterms.) For B ∞ = 0 the effective action is also invariantunder conformal transformations. The scalar is no longera propagating degree of freedom.For the realization of an UV-fixed point the anomalousdimension of the cosmon will be crucial. Indeed, for acanonical scaling of χ the “mass term” ∼ µ χ would spoilscale invariance for µ → ∞ . An anomalous dimension for µ → ∞ is realized if B ( χ/µ ) diverges for µ → ∞ with apower law, B = b (cid:18) µχ (cid:19) σ = (cid:18) mχ (cid:19) σ . (5)(We take χ = 0, negative χ being covered by the symmetry χ → − χ .) For σ = 2 the gravitational higher order invari-ants (3) (with constant C, D, E ) and the scalar kinetic termare then invariant under the scaling g µν → α g µν , χ → α − − σ χ. (6)At the UV-fixed point the effective action can be writtenin terms of a renormalized scalar field, χ R = b (cid:16) − σ (cid:17) − µ σ χ − σ , (7)as Γ UV = Z x √ g (cid:26) ∂ µ χ R ∂ µ χ R − CR + D (cid:18) R µν R µν − R (cid:19)(cid:27) . (8) For constant C and D scale invariance is manifest - it is therenormalized field that shows the standard scaling χ R → α − χ R . It is possible that C and D vanish at the fixedpoint, such that only a kinetic term for the renormalizedscalar field is left.The remaining terms for the cosmon potential and ∼ R read ∆Γ UV = Z x √ gc UV (cid:18) µ − R (cid:19) µ − σ − σ | χ R | − σ ,c UV = b − − σ (cid:12)(cid:12)(cid:12) − σ (cid:12)(cid:12)(cid:12) − σ . (9)The limit µ → ∞ and fixed χ corresponds to χ R /µ → σ <
2, and χ R /µ → −∞ for σ >
2. In sect. V wewill discuss that for σ > UV accounts for de-viations from the fixed point and can be neglected at the U V -fixed point. At the fixed point one finds a free mass-less scalar field coupled to higher order gravity. For theboundary case σ = 1 the renormalized scalar field has anon-vanishing self-interaction, with scale invariant poten-tial V ( χ R ) = 116 b χ R . (10)Also the limiting case σ = 2 can be associated with a fixedpoint - see sect. V. An UV-fixed point is therefore realizedfor σ ≥ . (11)Besides the cosmon-gravity part of the effective action(2), (3) we also have to specify the part for matter and ra-diation. We will take the standard model of particle physicsand assume that all renormalized dimensionless couplings(e.g. gauge couplings, Yukawa couplings, Higgs-boson selfinteraction), normalized at momenta ∼ µ , are functions of χ/µ that reach fixed constant values for µ → ∞ and µ → µ → ∞ , leaving only free particles at theUV-fixed point.We write the coefficient of the quadratic term in theHiggs potential as − ǫ H ( χ/µ ) χ . For an IR-fixed point thedimensionless coupling ǫ H goes to a (very small) constant.It actually becomes independent of µ for small enough ra-tios µ/χ , cf. app. A. No memory of the scale µ is then leftfor χ → ∞ - the Fermi scale is proportional to χ such thatthe charged lepton and quark masses as well as the gaugeboson masses are proportional to χ [5]. Consider next thestrong gauge coupling normalized at momenta given by theFermi scale, ¯ g s = g s ( Q = ǫ H χ ). It can only depend onthe ratio Q /χ = ǫ H . For ǫ H independent of µ also ¯ g s isindependent of µ . Therefore Λ QCD scales ∼ χ , such thathadron masses are ∼ χ as well.On the opposite end we may assume that the renor-malized coupling corresponding to ǫ H does not diverge for χ →
0. With masses ∼ χ all particles are then masslessat the UV-fixed point. Massless particles at the UV-fixedpoint are also realized for diverging ǫ H , provided the renor-malized masses scale with a positive power of χ .The same general picture applies for particles beyond thestandard model, in particular the sector of heavy singletswhich influence the neutrino masses by the seesaw [22–24]or cascade [25–28] mechanism. The only difference to thestandard model sector will be the relevant value of χ/µ for which the crossover between the two fixed points takesplace.The cosmology at the fixed points is not per se veryinteresting. For the past fixed point matter and radiationmay be negligible. The field equations for the cosmon-gravity system derived from the effective action (2), (3)admit for constant C and D the simple solution R µν = µ g µν , χ = 0 . (12)Another solution is simply flat Minkowski space, g µν = η µν , χ = 0 . (13)The cosmology for the future fixed point is again of asimple type. A scale invariant model that obtains by omit-ting in eq. (2) the potential µ χ has been proposed byFujii [29, 30]. After Weyl scaling it describes standard cos-mology plus a massless dilaton with derivative couplings.The dilaton settles to a fixed value after a short period ofinitial damping of its motion, and plays no role for the sub-sequent “late” cosmology of the present epoch [5]. Whileinteresting in its own right, such a scale invariant modelcannot describe dynamical dark energy or quintessence. Inour setting a cosmology similar to this model is reached forthe future fixed point.The interesting cosmological features of inflation and dy-namical dark energy are a consequence of small scaling vio-lations in the vicinity of the fixed points. Close to the pastfixed point the scale symmetry violating terms (“dilatationanomaly”) ∆Γ UV , cf. eq. (9), render the cosmological so-lution (12) unstable, such that any small value of χ slowlyincreases with increasing time t . This slow increase will beassociated with the almost scale invariant epoch of infla-tion. As χ grows large enough the crossover to the futurefixed point starts and inflation ends.The subsequent radiation and matter dominated epochsbelong already to the neighborhood of the standard modelfixed point. For this fixed point the dominant scaling vi-olation arises from the cosmon potential ∼ µ χ . Thiswill describe dynamical dark energy, according to an ap-proximate scaling solution with dark energy proportionalto the dominant radiation or matter component. Indeed,the ratio of the potential V = µ χ divided by the fourthpower of the effective Planck mass χ decreases ∼ µ /χ and reaches tiny values as χ moves to very large values inlate cosmology. Such a behavior amounts to a solution ofthe “cosmological constant problem”. At the second stepof the crossover an additional violation of dilatation sym-metry in the neutrino sector stops the scaling evolution ofthe cosmon.Realistic scaling solutions with a small fraction of earlydark energy have been extensively discussed [4, 5, 8] forthe case where B reaches for µ → B ( χ → ∞ ) = B ∞ = 4 α . (14) In the context of crossover gravity this would be the valueof B at the IR-fixed point. Observational bounds on earlydark energy restrict the allowed values to α &
10 [31–36]. (In view of possible degeneracies in the parameterspace of our model we take here a conservative bound.)While this setting is perfectly viable, we investigate in thepresent paper the possible alternative that B vanishes atthe IR-fixed point, B ∞ = 0. For the “conformal value” B = 0 the cosmon is no propagating degree of freedom.Furthermore, for B < B = 0 is a fixed point of the flow of B , and wewill assume that it is reached for χ → ∞ . As mentionedbefore, the IR-fixed point realizes conformal symmetry inthis case. For finite χ one has B >
4. Crossover
The crossover that leads to the end of inflation is relatedto the flow of the dimensionless function B ( χ/µ ). As a firstexample we take σ = 1 and discuss a one parameter flowequation µ ∂B∂µ = κB κB . (15)The approach to the fixed point at B = 0 is quadratic(vanishing anomalous dimension) µ∂ µ B = κB for B → , (16)while the approach to the fixed point B − = 0 involves ananomalous dimension µ∂ µ B − = − B − . (17)The finer details of the crossover will be less important.The UV-fixed point B − = 0 is approached for µ → ∞ or χ →
0. This fixed point is relevant for the infinite pastof our universe. The IR-fixed point B = 0 is approachedfor µ → χ → ∞ . It governs the infinite future. The so-lution of eq. (15) involves an integration constant c t whichdetermines the particular trajectory of the flow accordingto the implicit solution B − − κ ln B = κ (cid:20) ln (cid:18) χµ (cid:19) − c t (cid:21) = κ ln (cid:16) χm (cid:17) . (18)It is related to a mass scale m by dimensional transmu-tation, m = µ exp( c t ). The crossover between the twofixed points occurs in the region χ ≈ m and we will seethat this coincides with the end of inflation. Late cosmol-ogy corresponds to χ ≫ m , while primordial cosmology ischaracterized by χ ≪ m . With B ( χ ) determined by eq.(18) our model (2) contains two free dimensionless param-eters in the scalar-gravity sector, namely κ and c t . We willfind below that realistic cosmology can be obtained in theregion κ = 0 . , c t = 14 . (19)No tiny or huge dimensionless parameters appear in oursetting.The flow equation (3) is is only a particular example fora crossover between two fixed points for which B − or B vanish, respectively. For an arbitrary anomalous dimension σ it is generalized to an extended family of models, µ ∂B∂µ = κσB σ + κB , (20)with solution B − − κσ ln B = κ (cid:20) ln (cid:18) χµ (cid:19) − c t (cid:21) = κ ln (cid:16) χm (cid:17) . (21)Eqs. (15), (18) correspond to σ = 1, while eq. (20) indeedaccounts for an arbitrary anomalous dimension σ at theUV-fixed point, σ = lim µ →∞ µ∂ µ ln B. (22)The values of realistic parameters do not depend stronglyon σ . For σ = 3 one has c t ≈ χ/m relevantfor nucleosynthesis and later epochs the dimensionless cou-plings are already very close to their fixed point values,such that their dependence on χ can be neglected for thepurpose of cosmology. Similarly, we assume for these pe-riods that the masses of all particles except for neutrinoshave reached the scaling behavior m p ∼ χ appropriate forthe fixed point. With this simple assumption the severeobservational bounds on the time variation of fundamentalcouplings and apparent violations of the equivalence prin-ciple are obeyed [5].Neutrino masses also involve a sector of superheavy par-ticles by virtue of the seesaw or cascade mechanism. Theseparticles are part of the beyond standard model sector ofparticle physics. For this sector we postulate that thecrossover is happening in the region of χ/µ relevant forpresent cosmology, such that the present variation of theaverage neutrino mass with χ , ∂ ln m ν ∂ ln χ | today = 2˜ γ + 1 , (23)involves a parameter ˜ γ >
0. This parameter only mattersfor a rather recent cosmological epoch when neutrinos havebecome non-relativistic. It plays no role as long as neutri-nos are relativistic. Together with the present values forthe masses and couplings of particles, including some darkmatter candidate, the four parameters κ, σ, c t and ˜ γ willbe sufficient to describe a realistic cosmological sequence ofinflation, radiation- and matter-domination, as well as thepresent transition to a new dark energy dominated epoch.All four parameters are of the order one and no particularfine-tuning is needed. At present, it seems that our modelis compatible with all cosmological observations. We will see that the anomalous dimension σ is closely related tothe spectral index of the primordial fluctuations.The field equations derived from the effective action (2),together with the crossover of the kinetial (21) and thecrossover parameter in the neutrino sector ˜ γ (eq. (23)) arethe practical basis for computing our cosmological results.Besides the deeper motivation from quantum gravity ourmodel stands alone as a simple phenomenological descrip-tion of cosmology. III. Primordial cosmology and inflation
In this section we discuss the early cosmology of ourmodel. It is governed by the proximity of the ultravioletfixed point and describes an epoch of inflation. After abrief discussion of this epoch within the freeze frame ofvariable gravity, we perform a field transformation to theEinstein frame with a standard exponential form of the in-flaton potential. This is most suitable for a simple detaileddiscussion of the properties of primordial fluctuations. Forthe observable aspects of the inflationary epoch the higherorder curvature invariants play no role. We can thereforelimit the discussion to the effective action (2), for whichthe crossover is described by the kinetial (21).
1. Primordial cosmology
We begin with a brief discussion of primordial cosmol-ogy in the freeze frame. The field equations derived fromthe effective action (2), (3) are displayed and discussed forconstant C and D in the appendix C. For C > ∼ CR only affects the remote pastof the universe before observable density fluctuations leftthe inflationary horizon. We refer the discussion of the in-teresting properties of the infinite past, where the highercurvature terms play a role, to the appendices C and D.For our discussion of the “observable epoch” of inflationwe can omit the higher order curvature invariants (3). Asan example, we start with σ = 1 and approximate χ ≪ m, B = m/χ . The primordial epoch will correspond toan inflationary universe and we can neglect matter andradiation. The cosmon field equation obtains by variationof the effective action (2) and reads [4]¨ χ + (cid:18) H + 12 ˙ χχ (cid:19) ˙ χ = 2 µ χ m . (24)Here we have inserted the expression for R according tothe gravitational field equation. The Hubble parameter isgiven by H = s µ m ˙ χ χ − ˙ χχ . (25)The evolution of the mean value of the inflaton and ge-ometry until the end of inflation are described by the twoequations (24), (25).We may use dimensionless variables y = mt, w = χ/m , h = H/m, λ = µ /m , such that ∂ y w + (3 h + 12 ∂ y ln w ) ∂ y w = 2 λw ,h = r λ ∂ y w ) w − ∂ y ln w. (26)One finds an approximate solution that approaches a con-stant h for w → h = r λ − y c − y ) ,w = w ( y ) + ¯ w ( y c − y ) . (27)The function w ( y ) vanishes in the infinite past for y →−∞ , w = 12 r λ ( y c − y ) − , (28)and y c , ¯ w integration constants. Restoring dimensionsyields in leading order H = µ √ , χ = √ m µ ( t c − t ) . (29)We conclude that time can be continued in this approxi-mation to the infinite past, t → −∞ . In this limit geometryapproaches de Sitter space and the cosmon field vanishes.The limiting solution H = µ/ √ χ = 0 is unstable, how-ever. A small deviation χ increases with t according toeq. (29) or (27), and the Hubble parameter decreases. Pri-mordial cosmology describes an inflationary epoch. Thiswill end if the increase of χ or decrease of H becomes toofast. A quantitative estimate for the end of inflation willbe given later in the Einstein frame.A similar qualitative behavior extends to other values ofthe anomalous dimension σ . For our second example σ = 2we use B = m /χ , such that the field equations take theform ¨ χ + 3 H ˙ χ = 2 µ χ m ,H = s µ m ˙ χ χ − ˙ χχ . (30)The leading order solution becomes now H = µ √ , χ = 3 m √ µ ( t c − t ) − . (31)For σ > χ towards the infinitepast, cf. appendix C, eq. (C.10), χ ∼ ( t c − t ) σ . (32)We will see below that the solutions (29), (31) correspondto a standard inflationary scenario in the Einstein frame or “big bang frame”. Horizon crossing of the observableprimordial fluctuations occurs when χ/µ is already large, χ ≈ . · µ . For these values the relative contribution ofhigher order curvature invariants from eq. (3) is suppressedby a factor ∼ Cµ /χ , which is tiny for any moderate C .We can therefore indeed neglect such terms for the discus-sion of observable signals from inflation. Nevertheless, as χ becomes much smaller than µ for t → −∞ , the role ofthe higher order curvature invariants becomes important.We give more details of the behavior of cosmology in theinfinite past in appendix C. This includes the role of thehigher order curvature invariants. For constant C = 0 and σ < H = H is a free parameter.We will show next that our crossover model predicts arather large ratio between primordial tensor and scalar fluc-tuations. Since the value of the cosmon field χ (which playsthe role of the inflaton) equals the dynamical Planck mass,the Lyth bound [37, 38] plays no role in our setting of vari-able gravity [39].
2. Cosmon inflation
The association of primordial cosmology with an infla-tionary epoch is most easily understood in the Einsteinframe. Also a quantitative discussion of the generation ofprimordial density fluctuations and the end of inflation isbest done in this frame. With g ′ µν = χ M g µν , ϕ = 2 Mα ln (cid:18) χµ (cid:19) (33)the quantum effective action (2) readsΓ = Z x p g ′ (cid:26) − M R ′ + V ′ ( ϕ ) + 12 k ( ϕ ) ∂ µ ϕ∂ µ ϕ (cid:27) ,V ′ ( ϕ ) = M exp (cid:16) − αϕM (cid:17) . (34)We identify M = 2 . · GeV with the Planck mass andobserve that the cosmon potential V ′ decays exponentiallyto zero [5, 8]. The absence of an additional constant for ϕ → ∞ in the Einstein frame is a direct consequence of thevanishing ratio between potential and fourth power of thedynamical Planck mass, V /χ = µ /χ , for χ → ∞ in thefreeze frame.It has been advocated [3, 4, 40] that it is advantageousto use a field basis where the potential takes a fixed formwhile the detailed model information appears in the formof the coefficient of the kinetic term, the kinetial. Thereason is that the association of the value of the scalarfield with the value of the potential energy is universal fora standardized potential. This makes it easy to comparedifferent models. For our choice of a standard exponentialpotential the slow roll parameter ǫ and η reflect indeed verysimple properties of k ( ϕ ). We could choose α = 1 as far asinflation is concerned, but we prefer here a different valuein order to match the notation of quintessence potentialsfor late cosmology.The kinetial k is related to B by k = α B . (35)Since the parameter α appears only in the definition of ϕ , eq. (33), one is free to choose it at will and we couldindeed have set α = 1. Instead, we find it convenient toadopt a definition of α such that the field ϕ has a standardnormalization for the present cosmological epoch, k ( ϕ ) =1, or α = 4 B ( χ = M ) ≈ κ ln( M/m ) . (36)Typical values of α will be around ten or somewhat larger,see below. The normalization of ϕ and the precise value of α do not matter for the physics of inflation, however. For χ → B = (cid:18) mµ (cid:19) σ exp n − σαϕ M o . (37)A slow roll period for inflation is realized for large enough k . We consider here a general function B ( χ ) and specializeto eq. (21) later. The usual slow roll parameters ǫ and η obtain as [3, 4] ǫ = α k = 2 B , η = 1 B (4 − σ ) , σ = − ∂ ln B∂ ln χ . (38)Inflation ends when ǫ or | η | are of order one. We definethe end of inflation by the field value χ f determined by B ( χ f ) = 6, with ǫ f = 1 / , η f = 2 / − σ/
6. This is thevalue where the kinetic term in eq. (2) changes sign. In-flation is realized for a rather generic shape of the function B ( χ ). It is sufficient that B is large enough for small χ inorder to induce an epoch of slow roll, and that B − χ increases in order to end inflation.The definition of σ employed in the present section, givenby eq. (38), differs slightly from the preceding section. Inthe present section, σ is considered as a function of χ . Itagrees with the parameter σ in the preceding section for χ →
0. For the inflationary period this difference is minor(except possibly for the end of inflation), justifying the useof the same symbol.
3. Spectral index and tensor ratio of primordialfluctuations
The spectrum of primordial scalar density fluctuations ischaracterized by the spectral index n = 1 − ǫ + 2 η , whilethe relative amplitude of tensor fluctuations over scalarfluctuations reads r = 16 ǫ . Here ǫ and η have to be eval-uated for the value of χ at horizon crossing, N e -foldingsbefore the end of inflation. One finds the relations r = 32 B ( N ) , − n = r (cid:18) σ ( N ) (cid:19) . (39)We observe an interesting general relation between n and r . Horizon crossing occurs in the region χ ≪ m , B ≫ σ = 1, σ = 2 or σ = 3 predict1 − n = 3 r , r , r , (40) respectively. For a spectral index n = 0 .
97 this implies arather high amplitude, r = 0 . , . , . σ > σ < χ ( N )at horizon crossing and the number N of e -foldings beforethe end of inflation, N = 1 αM Z ϕ f ϕ dϕ ′ k ( ϕ ′ ) = 12 Z χ f χ ( N ) dχχ B ( χ ) . (41)In the range of interest B ( χ ) is typically a strongly de-creasing function. The integral is dominated by the regionaround χ ( N ) where we may approximate B = ( m/χ ) σ ( N ) .This relates B ( N ) = B (cid:0) χ ( N ) (cid:1) to N , N = B ( N ) − B ( χ f )2 σ ( N ) . (42)With B ( χ f ) = 6 one finds r = 16 N σ ( N ) + 3 (43)and n = 1 − σ ( N ) N σ ( N ) + 3 . (44)These two central formulae express both n and r in termsof σ ( N ) and N .For the particular model with σ = 1 one obtains B ( N ) =2 N + 6 and predicts r = 16 N + 3 , − n = 3 N + 3 . (45)while σ = 2 yields r = 162 N + 3 , − n = 42 N + 3 , (46)and σ = 3 results in r = 163 N + 3 , − n = 53 N + 3 . (47)We show the spectral index and the tensor ratio as a func-tion of σ for various N in Figs. 2, 3.We will see below that N depends only very mildly on σ . Its precise value shows some influence of the details ofthe entropy production after the end of inflation. A typicalvalue is N = 60. For a given N both n and r are uniquelydetermined by σ . More precisely, σ = σ ( N ) is defined interms of the function B ( χ ) by σ = − ∂ ln B∂ ln χ | B =2 σN +6 . (48)Thus only the logarithmic derivative of ln B at a particularvalue of B matters for the computation of n and r ! For N = 60 we display the values of n and r for three values of σ in table 1.0 σ n r σ, N = 60.FIG. 2: Tensor ratio r for primordial fluctuations as function ofthe anomalous dimension σ . The curves from top to down arefor N = 55 , ,
65. We also show the result for eq. (83) whichalmost coincides with N = 60.FIG. 3: Spectral index n for primordial fluctuations as functionof the anomalous dimension σ . The curves from top to downare for N = 65 , ,
55. We also show the result of eq. (83) whichalmost coincides with N = 60. We can use eqs. (43), (44) in order to relate r to 1 − n .Within the relation r − n − N + 1 − N + 1) (cid:18) − σ ( N ) (cid:19) (49)we can use the lowest order relation for σ ( N ), evaluatedfor N = 60, 3 σ ( N ) = 90(1 − n ) − . . (50)This yields r = 8 . − n ) − . . N − . (51)For n = 0 .
97 this predicts r = 0 . n =0 .
965 (0 . r = 0 .
15 (0 . r . . n close to one, n & . σ & .
5. We conclude that our model can be falsified by preci-sion observations of the CMB. Since inflation lasts for anextremely long time before horizon crossing of the observ-able fluctuations, perhaps even since the infinite past, thereseems to be no issue that memory of the initial conditionscould spoil its predictivity [44, 45]. If r and n can be estab-lished in accordance with the relation (51) this will consti-tute a measurement of the anomalous dimension σ . Hope-fully, this anomalous dimension is computable in quantumgravity, leading to a direct observational test.The relation (51) is approximately valid for a large classof inflationary models beyond our particular setting. Theslow roll parameters ǫ and η only involve the value and thederivative of the kinetial at the value of ϕ corresponding tohorizon crossing of the observable fluctuations. The rela-tion (51) follows qualitatively whenever the ϕ ′ -integral ineq. (41) is dominated by the large value of k at ϕ , whilethe decrease of k is well approximated by its first deriva-tive. Within eq. (42) the model-uncertainty can be castinto values of B ( χ f ) deviating from the value B ( χ f ) = 6for our setting.
4. Amplitude of primordial fluctuations
The amplitude of the primordial scalar fluctuations canbe related to the value of the potential at horizon crossingand the tensor to scalar ratio A = V out rM = 3 . · − , (52)where the last equation employs the observed amplitudeof the spectrum of CMB-anisotropies. This measurementdetermines the ratio mµ = χ ( N ) µ mχ ( N ) = M √ V out mχ ( N ) = 1 √A r mχ ( N ) . (53)We next employ the approximate form B = ( m/χ ) σ or mχ ( N ) = B ( N ) σ ( N ) = (cid:16) r (cid:17) − σ ( N ) , (54)such that mµ = 14 √ A B ( N ) + σ ( N ) = 2 σ ( N ) − (cid:0) N σ ( N ) + 3 (cid:1) + σ ( N ) A − . (55)For the particular model with σ = 1 one finds mµ = ( N + 3) √A = 1 . · (cid:18) N (cid:19) . (56)For the constant c t in eq. (18) one infers c t = ln (cid:18) mµ (cid:19) = 14 . . (57)(For the numerical value we have taken N = 60, see below.)Due to the exponential dependence on c t no very large1or small parameter is needed in order to obtain a smallfluctuation amplitude A = ( N + 3) e − c t . (58)The flow equation (15) generates the scale m by dimen-sional transmutation. The small amplitude A indicatesthat this scale is larger than the “intrinsic scale” µ . Thesituation is similar for other values of σ . For σ = 2 the ra-tio m/µ decreases by a factor 1 / √
30 as compared to σ = 1.For σ = 3 one has mµ = 2 / (cid:2) N + 1) (cid:3) A − / (59)and c t = 11 . . (60)We may turn this argument around and state thatcrossover models provide for a natural explanation of asmall fluctuation amplitude A . We can relate the dimen-sionless cosmon potential V /µ = χ /µ to a dimensionlessflow parameter by˜ µ = ln (cid:18) µχ (cid:19) = −
12 ln (cid:18) Vµ (cid:19) . (61)We have associated the scale m with the crossover value µ cr where the flow moves away from the behavior dictatedby the “past fixed point” for ˜ µ → ∞ , mµ = e − ˜ µ cr . (62)Different trajectories (solutions of the flow equations) canbe characterized by how close to the fixed point they arefor ˜ µ = 1. The larger m/µ , the closer a trajectory is to thefixed point. In view of the exponential behavior of eq. (62),already moderate negative values of ˜ µ cr are sufficient toinduce large values of m/µ , and therefore a small amplitude A ∼ ( µ/m ) ∼ e µ cr .We may also evaluate the dimensionless ratio Vχ = µ χ = e µ . (63)For χ = m this quantity measures the potential in units ofthe Planck mass at the crossover. For many of our modelsthe corresponding scale of the potential in the Einsteinframe V is of the order where spontaneous symmetrybreaking is expected in a grand unified theory. This sug-gests that the crossover could be associated with grandunified symmetry breaking.Finally, we may compare the value of χ ( N ) at horizoncrossing with m using eq. (54), x ( N ) = χ ( N ) m = (cid:16) r (cid:17) σ . (64) For all models one finds a small value x ( N ) ≪
1, justifyingthe approximation (5). On the other hand, we observe that χ ( N ) is much larger than µ , cf. eq. (53) χ ( N ) µ = 1 A r . (65)A simple picture arises. Horizon crossing happens when χ is already much larger than µ , but still smaller than m .Inflation ends when χ reaches m .
5. Horizon crossing
We finally need to evaluate the value of N for our typeof crossover models. We present here a detailed treatmentthat allows one to estimate where various uncertaintiescome from. Horizon crossing of a mode with comovingwave vector k occurs for k = a out H out = a in H in , (66)where a out or a in corresponds to the scale factor when themode leaves the horizon after inflation or enters again inthe more recent past.We use 1 = a out H out a in H in = a out a f a f a r a r a in H out H r H r H in , (67)with a f and a r the scale factors at the end of inflation andat a time when the universe begins to be dominated byradiation, respectively, and H r = H ( a r ). For a out /a f = e − N one finds the relation N = ln (cid:18) H r H in (cid:19) + ln (cid:18) H out H r (cid:19) − ln (cid:18) a in a r (cid:19) − ln (cid:18) a r a f (cid:19) . (68)Neglecting entropy production for photons for a > a r weuse a in /a r = T r /T in with T the photon temperature. Werelate T to the total energy density in radiation ρ r = 3 M H r = f r T r , ρ ( γ )in = 3 M H Ω ( γ )in = f in T , (69)with Ω ( γ ) the photon fraction of energy density and f r ( f in )the number of degrees of freedom in radiation (photons).These relations allow us to express a in /a r in terms of H in and H r . We further approximate H out /H f ≈ p V out /V f = χ f /χ ( N ), resulting in N = 12 ln (cid:18) H out H in (cid:19) −
14 ln f in f r Ω ( γ )in ! + 14 ln (cid:18) V out V f (cid:19) + ∆ N, ∆ N = 12 ln (cid:18) H f H r (cid:19) − ln (cid:18) a r a f (cid:19) . (70)We first evaluate N for modes that come into the hori-zon today, and subsequently extrapolate to larger k . Thedominant contribution is the first term ∼ ln( H out /H in ).We can relate H out to the tensor amplitude of the primor-dial fluctuations3 M H = V out = A r ( N ) M , (71)2and use 3 M H = ρ c = (2 · − eV) , which yields H out H in = 1 . √A r · . (72)With Ω ( γ )in = 5 · − and f r /f in = 100 one hasln( f r Ω ( γ )in /f in ) / − .
3. (Note that the neglected en-tropy production for photons can be incorporated intoa modification of the poorly known ratio f r /f in .) Withln A / − . N = 63 . r + 14 ln (cid:18) V out V f (cid:19) + ∆ N. (73)The two last terms involve the details of the epochs between a out and a f , or between a f and a r , respectively.For an estimate of V f we employ B ( χ f ) = 6 and eq.(21), ln (cid:16) χ f m (cid:17) = 16 κ − ln 6 σ , V f M = µ χ f . (74)With eq. (54) one finds V out V f = χ f χ ( N ) = exp (cid:18) κ (cid:19) (cid:18) r (cid:19) − σ , (75)or N = 63 . κ + 0 . σ + 14 (cid:18) − σ (cid:19) ln r + ∆ N, (76)with r depending on N and σ according to eq. (43). In-serting N ≈
65 in the subleading term ∼ ln r , and κ = (see below) yields for σ = 1( σ = 2) N = 65 . .
3) + ∆ N. (77)The remaining piece ∆ N reflects the details of entropyproduction between the end of inflation ( a f ) and the be-ginning of the radiation dominated universe ( a r ). We mayparametrize this epoch by two parameters, the number of e -foldings N fr for the duration of this period N fr = ln a r a f , (78)and the averaged equation of state ¯ w which governs theevolution of the total energy density, ∂ t ρ = − H (1 + ¯ w ) ρ. (79)With ρ = ¯ ρa − w ) = 3 M H , Ha ∼ a − w , (80)one finds ∆ N = 3 ¯ w − N fr . (81)For a fast entropy production N fr is of the order one. Theparameter ¯ w is a suitable average of a function w ( a ) that starts close to w ( a f ) ≈ − a = a f , may then be givenfor a period of domination of scalar kinetic energy, w ( a ) ≈
1, and finally end with w ( a r ) ≈ /
3. For not too large N fr and ¯ w close to 1 / N , and wewill use this approximation in the following.We may finally extrapolate to modes with present wave-length smaller than the horizon. As compared to N thedominant correction factor is N = N + δN,δN = ln (cid:18) H a H in a in (cid:19) = ln k k = − ln L L , (82)with k and L the wave number or wave length of the mode,and index zero denoting the ones corresponding to thepresent horizon ( L ≈ M pc ). In the range where pri-mordial gravitational waves may be detected ( k/k ≈ δN ≈ − .
4, such that a reasonable overall estimateis N ≈ − . σ − . (83)We can neglect the σ -dependence of N and use N = 60.Up to small calculable corrections for σ = 2 this entails thepredictions r = 0 . σn = 1 − . σ · (cid:18) σ − (cid:19) . (84)In particular, one obtains for σ = 2 r = 0 . , n = 0 . , (85)in accordance with the prediction in ref. [46] (see also refs.[47–49]). For σ = 1 one finds r = 0 . , n = 0 . N on the wave number k thespectral index and tensor amplitude depend on k accordingto ∂r∂ ln k = − ∂r∂N = r (cid:18) ∂σ∂ ln N (cid:19) , (86) ∂n∂ ln k = − ∂n∂N = − r (1 − n )16 (cid:18) ∂σ∂ ln N (cid:18) − σ − N (cid:19)(cid:19) . Since | ∂σ/∂ ln N | is typically of the order one or smallerthe running of the spectral index is very slow. Indeed, forour model σ changes over 60 e-foldings only from σ ( N ) to σ f = − ∂ ln B∂ ln χ | B =6 ≈ κB ≈ , (87)and this change occurs towards the end of inflation. Weconclude that our UV-fixed point scenario provides for arather simple and predictive model of inflation. IV. Late cosmology and dark energy
The crossover in the kinetial K ( χ ) = B ( χ ) − ǫ H could be much larger than the tiny value for the standardmodel fixed point. A fluctuating Higgs doublet or a sim-ilar field related to spontaneous symmetry breaking in agrand unified setting could play a major role for the heat-ing [3]. More precisely, a χ -dependence of ǫ H results inthe Einstein frame in an effective coupling between thecosmon and the Higgs doublet. This generalizes to other χ -dependent dimensionless couplings. In a grand unifiedtheory the heating period may be associated with the on-set of spontaneous symmetry breaking of the GUT-gaugegroup. Rather generically, the χ -dependence of couplingsis large precisely in the crossover region. Thus at the end ofinflation the cosmon coupling to other particles is large inthe Einstein frame, in contrast to the tiny couplings closeto the standard model fixed point. The large couplingsprovide for rather efficient heating mechanisms.After the heating and entropy production have occurredthe universe enters its “late epoch”, beginning with radia-tion domination. The late universe is characterized by theapproach to the future fixed point. During the radiationand matter dominated periods this approach is slow, asaccounted for by the (approximate) standard model fixedpoint, recall Fig. 1.In the freeze frame the particle masses increase with in-creasing χ , while the universe shrinks, in contrast to theusual big bang picture [46]. (For early cosmological modelswith varying particle masses see. refs. [50–52].) Indeed,only the dimensionless ratio of the distance between galax-ies divided by the atom radius is observable [53–56]. Theoverall picture of late cosmology in the freeze frame hasbeen described in detail in ref. [4] for the case where thekinetial takes a constant value K ∞ = B ∞ − H = bµ ,while particle masses increase exponentially, χ ∼ exp( cµt ).The characteristic time scale for both epochs is given by µ − = 10 yr, such that the evolution is always very slow.Temperature increases T ∼ √ χ due to the shrinking ofthe universe. Particle masses m p ∼ χ increase even faster,however, such that the relevant ratio T /m p decreases as inthe usual big bang picture.In the present crossover model B depends only mildlyon χ for χ ≫ m , B = 1 κ ln (cid:0) χm (cid:1) . (88)The cosmology with constant B ∞ = B ( χ → ∞ ) = 4 /α istherefore a good approximation. A priori, both the behav-ior (88) and a small fixed value of B ∞ are perfectly viablecandidates for realistic late cosmology. In the present pa-per we supplement the earlier discussion with constant B ∞ by a quantitative investigation of a slowly varying B ( χ ) ac-cording to eq. (88). We employ the Einstein frame in order to facilitate the embedding of this model in standard sce-narios of quintessence.
1. Late cosmology in the Einstein frame
In the Einstein frame (34) the kinetial is given for latecosmology by k = M α κ ( ϕ − ¯ ϕ ) = (cid:18) κ ( ϕ − ϕ ) M α (cid:19) − , (89)with ϕ the present value of the cosmon ϕ = 2 Mα ln Mµ , ¯ ϕ = 2 Mα ln (cid:18) mµ (cid:19) . (90)With the exponential potential (34), this is a typical modelof dynamical dark energy. (The divergence for ϕ → ¯ ϕ is outside the vicinity of the approximation.) Except forneutrinos the standard model particles and dark matter donot couple to ϕ . The cosmon field equation k ( ¨ ϕ +3 H ˙ ϕ )+ 12 ∂k ∂ϕ ˙ ϕ = αM exp (cid:16) − αϕM (cid:17) + βM ( ρ ν − p ν )(91)involves, however, the cosmon-neutrino coupling β ( ϕ ) = − M ∂ ln m ν ( ϕ ) ∂ϕ , (92)with m ν the ϕ -dependent average neutrino mass. The cou-pling β is large only in the range of ϕ which corresponds tothe second step of the crossover (CR2 in Fig. 1.) It playsno role as long as neutrinos are relativistic.The Hubble parameter obeys H = ρ M , ρ = ρ h + ρ r + ρ m + ρ ν , (93)with ρ h = V + k ϕ , p h = − V + k ϕ , (94)and ρ r,m,ν the energy densities of radiation, matter andneutrinos, respectively. While ρ r and ρ m obey the usualconservation equations, ˙ ρ r = − Hρ r , ˙ ρ m = − Hρ m , theneutrinos exchange energy momentum with the cosmondue to the variable mass˙ ρ ν + 3 H ( ρ ν + p ν ) = − βM ( ρ ν − p ν ) ˙ ϕ, ˙ ρ h + 3 H ( ρ h + p h ) = βM ( ρ ν − p ν ) ˙ ϕ. (95)(The second equation follows from eqs. (91), (94).)We may follow the evolution in terms of y = ln a + y instead of time [7, 57, 58], ∂ y ln V = − αM ∂ y ϕ = − α r h − Ω V ) k ,∂ y ln ρ h = − (cid:18) − Ω V Ω h (cid:19) + ˜ γ (1 − w ν ) Ω ν Ω h ∂ y ln V,∂ y ln ρ ν = − w ν ) − ˜ γ (1 − w ν ) ∂ y ln V,∂ y ln ρ r = − , ∂ y ln ρ m = − , (96)4with Ω V = V /ρ, Ω r,m,ν,h = ρ r,m,ν,h /ρ, w ν = p ν /ρ ν . Theparameter ˜ γ = − β/α (97)may depend on ϕ . Evaluated at the present value of ϕ it isthe same as in eq. (23). It will determine the precise timingof the crossover to dark energy domination. The system ofdifferential equations (96) can be solved numerically [7].As long as neutrinos are relativistic one has w ν = 1 / ∼ ˜ γ can be neglected. For the radiationdominated epoch we can neglect ρ m and incorporate ρ ν into ρ r , ρ r = ¯ ρ r M exp( − y ). For the matter domi-nated period neutrinos can be neglected as long as theyare relativistic, similar to radiation. We only need to keep ρ m = ¯ ρ m M exp( − y ). We may combine the discussion ofthese periods by taking ρ d = ¯ ρM exp( − ny ) for the energydensity of all other components except the cosmon, with n = 4(3) for radiation (matter) domination.The last epoch in the cosmological evolution starts whenneutrinos become non-relativistic. The terms proportionalto the cosmon-neutrino couplings β in eq. (95) can nolonger be neglected. For large enough ˜ γ they stop effec-tively the further change of ϕ , such that V ′ ( ϕ ) acts like acosmological constant. This scenario of “growing neutrinoquintessence” [6, 7, 59–62] relates the present dark energydensity to the average neutrino mass ρ h ( t ) = 1 . (cid:18) ˜ γm ν ( t )eV (cid:19) − eV . (98)(Here ˜ γ is evaluated today.) A realistic present dark energyfraction Ω h ( t ) ≈ . γm ν ( t ) = 6 . . (99)This relation remains valid with good accuracy even inpresence of the large scale non-linear neutrino lumps thatform and dissolve periodically after redshift z ≈
2. Approximate analytic solution
For a constant kinetial k = 1 one finds for radiationor matter domination the standard scaling (tracker) solu-tion for quintessence with an exponential potential [5]. Forslowly varying k ( ϕ ) according to eq. (89) we may thereforeuse the approximation of a solution in the vicinity of thescaling solution. The difference between the cosmon fieldaccording to a model with ϕ -dependent kinetial on oneside, and the scaling solution on the other side, is denotedby M δ ( y ). We will derive next an approximate analyticsolution for δ ( y ). The resulting time evolution of the earlydark energy fraction is found asΩ h = nB ( χ )4 , (100)instead of Ω h = nB ∞ / B ( χ ) changes only very mildly for recent cosmology theresults [7] of investigations with constant B ∞ continue tobe a very good approximation. For the evolution equations for ϕ and ln ρ h we make theansatz ρ h = f ( ϕ ) ρ d , ϕ = M (cid:16) nyα + δ ( y ) (cid:17) , (101)such that ∂ y ln f = n − f ¯ ρ exp( − αδ ) , (102) ∂ y δ = − nα + s fk (1 + f ) (cid:18) − f ¯ ρ exp( − αδ ) (cid:19) . For constant k one recovers the scaling solution [5, 8, 64]with a constant fraction of early dark energy Ω e , δ =0 , ∂ y f = 0,¯ ρ = 6(6 − n ) f , Ω e = f f = nk α . (103)For a smooth enough ϕ -dependence of k we thereforeexpect a behavior close to this scaling solution. We employ1 f = (cid:16) − n (cid:17) ¯ ρ exp (cid:2) − αζ ( y ) (cid:3) (104)and find ∂ y ζ = 6 − nα (cid:16) exp (cid:2) − α ( δ + ζ ) (cid:3) − (cid:17) , (105) ∂ y δ = − nα + r n Ω h k s (cid:18) n − (cid:19) (cid:16) − exp (cid:2) − α ( δ + ζ ) (cid:3)(cid:17) , with Ω h ( y ) = f ( y )1 + f ( y ) . (106)For the variables∆ = (cid:18) n − (cid:19) (cid:16) − exp (cid:2) − α ( δ + ζ ) (cid:3)(cid:17) ,u = 1 − α Ω h nk (107)one obtains ∂ y ∆ = (cid:2) − n (1 + ∆) (cid:3)(cid:0)p (1 + u )(1 + ∆) − − ∆ (cid:1) ,∂ y u = (1 + u ) (cid:26) M ∂ ln k ∂ϕ (cid:16) nα + ∂ y δ (cid:17) − α Ω h f ∂ y ζ (cid:19) . (108)At this point we assume that ( nM/α ) ∂ ln k /∂ϕ is small.We can then expand in small ∆ and u , ∂ y ∆ = n −
62 (∆ − u ) ,∂ y u = nMα ∂ ln k ∂ϕ − n Ω h (1 − Ω h )∆ . (109)5With ∂ ln k /∂ϕ and Ω h varying slowly the solution ap-proaches the particular approximate solution¯∆ = ¯ u = Mα Ω h (1 − Ω h ) ∂ ln k ∂ϕ . (110)Indeed, if we neglect the y -dependence of ¯∆ and ¯ u one hasfor u ′ = u − ¯ u, ∆ ′ = ∆ − ¯∆ the linear evolution ∂ y (cid:18) ∆ ′ u ′ (cid:19) = A (cid:18) ∆ ′ u ′ (cid:19) ,A = n − , − − n Ω(1 − Ω) n − , ! . (111)The eigenvalues of the stability matrix A are both nega-tive, implying an exponential decrease of ∆ ′ and u ′ as y increases. We conclude that cosmology approaches a solu-tion with non-vanishing early dark-energy fraction decreas-ing with decreasing k ,Ω h = nk α (1 − ¯ u ) . (112)Solving eqs. (112),(110) for Ω h we end with a general for-mula for slowly varying k ( ϕ ),Ω h ≈ nk α − (cid:18) − nk α (cid:19) − Mα ∂ ln k ∂ϕ . (113)Here we recall that for generic models of quintessence theformulation with exponential potential (34) and possiblyvarying kinetial k ( ϕ ) can be obtained by an appropriaterescaling of the scalar field.Let us now turn to our model with k = M α/ (cid:0) κ ( ϕ − ¯ ϕ ) (cid:1) and ∂ ln k ∂ϕ = − ϕ − ¯ ϕ = − κk M α . (114)One has ¯ u = − κn (1 − Ω h ) , (115)and for Ω h ≪ κ ≪ u ≪
1. In turn the early dark energy fractionΩ h = n κ Mα ( ϕ − ¯ ϕ ) = n κ ln (cid:16) χ m (cid:17) = nB ( χ )4 (116)decreases logarithmically for increasing χ .
3. Bounds on parameters
Besides the determination of ˜ γm ν ( t ) by a measurementof the present dark energy fraction (99) we can use boundson early dark energy for an estimate of the parameter κ .For nucleosynthesis the dimensionless ratio V /χ = µ /χ = V ′ /M is of the order (MeV) / (10 GeV) ≈ − . With m /µ ≈ this implies ln( χ/m ) ≈ ( ns ) h ≈ κ . (117) If we require κ < / u thisyields a dark energy fraction larger than 2% which couldbe detectable in the future [8, 65, 66].For the present epoch one has V ′ ( ϕ ) M = µ M = (cid:18) · − eV2 . · GeV (cid:19) = (0 . · − ) . (118)This sets the scale of our model µ = 1 . · − eV , (119)similar to the present value of the Hubble parameter. Withln( M/µ ) ≈ .
55, ln( m/µ ) ≈ . M/m ) =122 . α = 490 κ. (120)Interestingly, for κ < / α, α < . k haschanged only little k − ( z ) = 1 − κα ln(1 + z ) , (121)such that at last scattering the relation Ω h ≈ /α & / lsh = 3490 κ. (122)For κ < . lsh ≥ . . (123)This is at the borderline of a possible detection with presentobservations [31–36, 67]. One therefore infers the bound κ & .
5. We may take a value κ = 0 . u .Formally, we may combine eqs. (112), (115), (120) inorder to obtain for k ≈ , n ≈ , Ω lsh ≪ lsh = 3490 κ + 1245 . (124)This would imply a minimal value for Ω lsh , close to the lowerbound quoted in ref. [36]. (We recall, however, that theobservational bound may change if the other ingredients ofour model are included in the parameter estimation.) Theminimum is reached, however, only for large κ for whichour approximation no longer holds. We do not expect aqualitative change for somewhat larger values of κ . Whilethe analytical estimate becomes inaccurate for κ & .
5, anumerical solution can be extended easily to larger κ . Itwill be interesting to see if a saturation with a minimalvalue of Ω lsh , as suggested by eq. (124), takes place forincreasing κ . A lower bound on Ω lsh would make the presentmodel distinguishable from Λ CDM where Ω lsh is practicallyzero.6In summary, the late cosmology of our model resem-bles closely growing neutrino quintessence with a variablecosmon-neutrino coupling β [7]. The interesting new fea-tures are an explanation of a large effective value for α interms of the approach to the IR-fixed point, and the associ-ation of large positive ˜ γ = − β/α with a crossover affectingthe neutrino masses in the recent and present cosmologicalepoch. V. Ultraviolet fixed point
The fixed point that is relevant for the infinite past t → −∞ (“past fixed point”) corresponds to χ →
0. It ischaracterized by an anomalous dimension σ that appearsin the scalar kinetic term. (In the language of universalcritical exponents σ corresponds to − η ). The approachto the fixed point corresponds to B − →
0, with limitingbehavior of the flow equation (20) given by µ∂ µ B − = − σB − . (125)At the fixed point scale symmetry is exact and not spon-taneously broken. With all particle masses vanishing for χ →
1. Renormalized scalar field
The solution of eq. (125), B = (cid:18) mχ (cid:19) σ , (126)contains an explicit mass scale m , in addition to the massscale µ . (The ratio m/µ can be considered as a dimen-sionless coupling that specifies B besides the dimensionlessparameter σ .) For σ = 2 one can absorb m in the definitionof a renormalized field χ R = (cid:16) − σ (cid:17) − B χ = (cid:16) − σ (cid:17) − m σ χ − σ . (127)For σ < χ → χ R → σ < χ R → −∞ . In terms ofthe renormalized field the effective action contains a scalarkinetic term with standard normalizationΓ = Z x √ g (cid:26) ∂ µ χ R ∂ µ χ R + e σ | χ R | − σ (cid:18) µ − R (cid:19)(cid:27) ,e σ = (cid:12)(cid:12)(cid:12) − σ (cid:12)(cid:12)(cid:12) − σ m − σ − σ . (128)For the particular case σ = 1 this yields ( λ = µ /m )Γ = Z x √ g (cid:26) ∂ µ χ R ∂ µ χ R + λ χ R − χ R m R (cid:27) . (129)The last term ∼ R vanishes for χ R /m →
0, such that no mass scale remains in this limit. We may define a dimen-sionless coupling˜ λ R = Vχ R = µ χ χ R = µ m (cid:12)(cid:12)(cid:12) σ − (cid:12)(cid:12)(cid:12) − σ − (cid:18) | χ R | m (cid:19) − σ − σ − . (130)For σ = 1 one has ˜ λ R = λ/
16. For σ > σ < | χ R | →
0. In this limit ˜ λ R goes to zero. The potentialterm becomes subleading and can be neglected in the UV-limit. With both V and χ R neglected in the UV-limit theeffective action contains indeed no mass scale. For σ > | χ R | → ∞ . Again ˜ λ R vanishes in this limit and Γ does not involve a mass scale inthe UV-limit. This behavior demonstrates scale invarianceat the fixed point very explicitly. For the boundary case σ = 2 one finds a logarithmic dependence of χ R on χχ R = m ln (cid:16) χm (cid:17) , χ = m exp (cid:18) χ R m (cid:19) . (131)The fixed point is now realized for χ R → −∞ where both V /χ R and χ R/χ R vanish. We conclude that for the wholerange σ ≥ σ . While the renormalized scalar field χ R scalesproportional to mass, the original scalar field χ scales ∼ mass / (2 − σ ) . (For the example σ = 1 one finds a scaling of χ ∼ mass ). Thus the effective action becomes invariantunder the scaling g µν → α g µν , χ → α − − σ χ , χ R → α χ R . (132)An interesting particular case is σ = 3 where χ scaleswith the same factor as the metric. The term ∼ R in-volves a scale symmetry violation which vanishes in thelimit χ/m →
0. It characterizes a relevant parameter forthe deviation from the fixed point as χ increases. For σ = 1the term ∼ µ χ is invariant under the scaling (132), as eas-ily visible in eq. (129). This situation changes for σ > σ = 3 / χ scales ∼ mass andΓ = Z x √ g (cid:26) ∂ µ χ R ∂ µ χ R + 2 − µ χ R m − − χ R m R (cid:27) . (133)The two last terms vanish in the limit χ/m → σ = 3,Γ = Z x √ g (cid:26) ∂ µ χ R ∂ µ χ R + 16 m χ R (cid:18) µ − R (cid:19)(cid:27) , (134)where χ/m → χ R → −∞ . At the fixedpoint we are left for both examples with a free masslessscalar field. This simplicity makes the existence of such afixed point rather plausible.7We observe that for σ < e σ decreasesless than ∼ m − for m → ∞ . As a consequence the dimen-sionless quantity ˜ λ R = V /χ R diverges for χ/m →
0. Nofixed point is obtained in this case.At the fixed point the term ∼ R vanishes. However, onemay expect the presence of higher order invariants, as givenby eq. (3). Such terms are scale invariant and thereforecompatible with dilatation symmetry if the dimensionlessquantities C and D are constant. (Slowly running C and D would be considered as marginal parameters for deviationsfrom the fixed point.) In the limit χ → β -functions closeto their zeros. This will determine the coupling ∼ χ R and, for σ >
1, the term ∼ µ χ , as well as the flow ofcouplings of the standard model of particle physics. Weemphasize that a fixed point with the simple effective ac-tion (8) requires a substantial anomalous dimension σ >
2. Gauge hierarchy
We will next address possible interesting consequencesof an UV-fixed point for particle physics, in particularthe gauge hierarchy problem. This concerns the possi-bility that the effective coupling ǫ H between the Higgs-doublet and the cosmon, which determines the Fermi scale,is driven to very small values by its flow in the vicinity ofthe UV-fixed point.The gauge hierarchy is related to the small value of theeffective coupling ǫ H which appears in the quantum effec-tive potential for the Higgs doublet ˜ h [3, 4],˜ V h = 12 λ h ( χ/µ )(˜ h † ˜ h − ǫ h ( χ/µ ) χ ) , ǫ H = λ h ǫ h . (135)For the present range of χ the function ǫ h ( χ/µ ) must be(almost) independent of µ and have reached a very smallvalue ǫ h ( χ/µ = M/µ ) = 5 · − . Besides their dependenceon χ/µ the functions λ h and ǫ H also depend on ˆ h † ˜ h/χ .This latter dependence is described by the standard model β -functions. The running of ǫ H with ˆ h † ˜ h/χ is given bya perturbatively small anomalous dimension [69, 70]. Weneglect this small effect and use ǫ H ≈ − independentlyof ˜ h † ˜ h/χ . (A small value of ǫ H for a Higgs field value ofthe order of the dynamical Planck mass χ remains small fora Higgs field value equal to the Fermi scale. This propertyreflects the (almost) second order character of the elec-troweak phase transition - the associated effective scale in-variance of the non-gravitational physics protects a smallvalue of the Higgs mass term [69–76].) We explore hereif the small value of ǫ H can be caused by the running of ǫ H near the UV-fixed point, before it is stopped at thecrossover for χ ≈ m . In order to understand this issue we first consider theeffective renormalized quartic coupling for the cosmon for χ ≪ m , λ R = 124 ∂ V∂χ R , V = µ e σ ( µ ) | χ R | − σ , (136)which differs from ˜ λ R in eq. (130) only by a multiplicativeconstant. From λ R ∼ (cid:18) | χ R | µ (cid:19) σ − − σ ∼ (cid:18) χ µ (cid:19) σ − (137)we extract the flow equation µ∂ µ λ R | χ = − σ − λ R = A λ λ R . (138)For σ > A λ < λ R is asymptotically free in the ultraviolet. We observe thatthe anomalous dimension A λ can be quite large. A giventrajectory (model) can be specified by the value of λ R at χ/µ = 1. This is typically a rather small value, corre-sponding to the close vicinity to the fixed point. For largervalues of χ/µ the renormalized coupling λ R increases.We next turn to the cosmon-Higgs coupling that we de-fine as ǫ H = − ∂ ˜ V h ∂ ( χ ) ∂ (˜ h † ˜ h ) . (139)The corresponding renormalized coupling ǫ R = − ∂ ˜ V h ∂ ( χ R ) ∂ ( h † R h R ) (140)involves χ R and the renormalized Higgs doublet h R . Using B = ( χ/m ) σ , χ R /χ = √ B/ (cid:0) − σ (cid:1) , one has ∂ ( χ R ) ∂ ( χ ) = B − σ . (141)Similarly, the kinetic coefficient B h of the Higgs doubletmay depend on µ in the vicinity of the UV-fixed point,resulting in ∂ ( h † R h R ) ∂ (˜ h † ˜ h ) = B h − σ h , µ∂ µ ln B h = σ h . (142)This relates ǫ R and ǫ H ǫ H = BB h (cid:0) − σ (cid:1) (cid:0) − σ h (cid:1) ǫ R . (143)Let us now assume that ǫ R is asymptotically free in theUV, similar to λ R , µ∂ µ ǫ R = A ǫ ǫ R , A ǫ < . (144)This results in a flow of ǫ H according to µ∂ µ ǫ H = ( A ǫ + σ + σ h ) ǫ H = σ ǫ ǫ H . (145)8(We take σ and σ h approximately constant here.) While ǫ R decreases with increasing µ, ǫ H can increase if the sum σ + σ h overwhelms the negative contribution A ǫ such that σ ǫ > ǫ H ∼ (cid:18) µχ (cid:19) σ ǫ . (146)Turned around, ǫ H will then decrease for increasing χ andfixed µ .The behavior (146) is valid only for the vicinity of theUV-fixed point for χ . m . For the vicinity of the standardmodel-fixed point, χ ≫ m , we assume that ǫ H reachesrapidly its constant fixed point value. (Formally σ ǫ ≈ χ ≫ m .) Specifying the trajectory at a given ratio χ in /µ , ǫ in = ǫ H ( χ in /µ ), the value of ǫ H for χ ≫ m is reducedby a factor ǫ H ≈ (cid:16) χ in m (cid:17) σ ǫ ǫ in . (147)This factor could explain the gauge hierarchy. For ǫ in ofthe order one one needs χ in m ≈ − σǫ . (148)For example, for χ in = µ a value σ ǫ ≈ − ǫ H between χ = µ and χ = m by around 30 orders of magnitude. This would relate thesmallness of the ratio Fermi scale/Planck mass and thesmall amplitude of primordial density fluctuations, cf. eq.(55), h h i M ∼ A σ ǫ / . (149)(For χ in ≪ µ smaller values of σ ǫ would be sufficient toachieve the suppression factor needed for the gauge hierar-chy.)The possible emergence of a gauge hierarchy, expressedby the tiny coupling ǫ H ( χ ≫ m ) ≈ − , can be viewedfrom different perspectives. While ǫ H should be approxi-mately constant for χ ≫ m , nothing prevents an increaseof ǫ H for χ ≪ m , such that values of the order one can bereached for small enough χ . The increase of ǫ H towardsthe UV-fixed point remains compatible with an asymptoti-cally free renormalizable coupling ǫ R . For sufficiently small χ all asymptotically free renormalized couplings are verysmall. If the anomalous dimension | A ǫ | for the coupling ǫ R is smaller than the corresponding one for other couplingsthe coupling ǫ R still remains small at the crossover scalewhere the flow effectively stops and ǫ R roughly equals ǫ H .The coupling ǫ H measures the distance from the elec-troweak phase transition which is of second order (up tosmall QCD-effects). This guarantees that its flow vanishesfor ǫ H = 0. Such a setting generalizes to a large class ofmodels, including grand unified models. Then ǫ H measuresthe distance from the hyperface in coupling constant spacecorresponding the phase transition. While the location ofthis hypersurface may be complicated in a given basis for the couplings (often associated with a “fine tuning prob-lem”) the general structure of the flow equation for ǫ H remains the same [77].The two steps in the flow of ǫ H , first a fast decrease for χ ≪ m and then an almost constant behavior for χ ≪ m ,would realize an old idea for a possible explanation of thegauge hierarchy [69]. The necessary large values of anoma-lous dimensions are often found in the gravitational con-tribution to the flow [12–14, 78]. In our scenario σ has tobe large in order to realize an UV-fixed point. Then also σ ǫ will typically have a large value, unless some particularcancellation occurs in eq. (145). Without an explicit com-putation of the µ -flow equation our discussion remains aneducated guess. It clearly shows, however, that an ultravio-let fixed point with large anomalous dimensions could playan important role for the gauge hierarchy problem. Thisalso applies for a possible understanding of the value ofthe Higgs boson mass. If the flow of the quartic Higgs cou-pling λ h ( χ/µ ) exhibits a large positive anomalous dimen-sion the “asymptotic safety scenario for the Higgs bosonmass” is realized [79], which has led to a predicted value m h ≈ C may be marginal. Graviton fluctua-tions could be responsible for large anomalous dimensions. VI. Field relativity
Once quantum fluctuations are included on the level ofthe quantum effective action the corresponding field equa-tions can be solved with arbitrary field variables. Valuesand correlations of physical observables are independent ofthe choice of fields used to describe them [53]. This exactproperty may be called “field relativity” [46]. Indeed, ob-servables are expressed as functionals of fields. Again, theycan be written in terms of arbitrary field variables. In gen-eral, the specific functional expression for a given physicalobservable will be changed under a change of field variables(see refs. [53, 68] for the transformation of some quanti-ties relevant for cosmology as temperature or proper time.)We stress that only dimensionless quantities can be physi-cal observables [53]. Different choices of field variables arecalled different frames. A well known example for a frametransformation is the Weyl transformation from the Jor-dan to the Einstein frame [80, 81] that we have employedin sect. III.In the present section we employ frame transformationsfor several different purposes. We first show that a verylarge class of coupled scalar-gravity models can be broughtto the form (2), with B ( χ/µ ) the only free function. Typi-cally this holds if the field equations contain no more thantwo derivatives and the scalar potential is monotonic. A9formal treatment of a large class of such models, includ-ing the ones of the Horndeski type [82], can be found inref. [83]. Our discussion of a crossover between a past andfuture fixed point can therefore be carried over to a largeclass of models.We have described the crossover as a “kinetial crossover”where the relevant information is encoded in the scalar ki-netic term, i.e. the function B ( χ/µ ). Field transformationscan be used to express the same physics as a “potentialcrossover” [84], where the information is now containedin the shape of the scalar potential V ( χ/µ ), while the ki-netic term has a standard normalization. We also presenta “primordial flat frame” for which the cosmological solu-tion approaches flat space in the infinite past for modelswithout higher order curvature invariants. Finally, we castthe effective action into the form of a free scalar field cou-pled to gravity. While the kinetic term is standard and thepotential quadratic, the crossover information is now con-tained in a χ -dependent function multiplying the curvaturescalar. Having at hand the formulation of the ultravioletand infrared fixed points in different frames may facilitatethe search for such fixed points in a genuine quantum grav-ity calculation.We omit in this section higher order curvature invariantsas in eq. (3). They would have to be transformed appro-priately under field transformations. This section thereforedeals with various expressions for the quantum effective ac-tion encoded in eq. (2).
1. Field transformations within Jordan frames
We will call “Jordan frames” the choice of fields for whichthe curvature scalar in the effective action (2) is multipliedby χ . We allow for a general potential V ( χ ) instead of µ χ in eq. (2). In contrast, the “Einstein frame” or “bigbang frame” (34) has a constant coefficient M in frontof the curvature scalar. The Einstein frame is unique, upto a choice of the scalar field ϕ which may be replaced by χ or a field σ with standard normalization of the kineticterm. The Jordan frames, however, are not yet uniquelyfixed, since there exist field transformations keeping theterm ∼ χ R invariant, while changing V ( χ ) and B ( χ ). Theparticular choice of fields where V ( χ → ∞ ) = µ χ will becalled “freeze frame”. We may parametrize the Jordanframes by two dimensionless functions B ( χ/µ ) and v (cid:18) χµ (cid:19) = V ( χ ) χ . (150)Indeed, the most general quantum effective action with nomore than two derivatives takes in the Jordan frame theformΓ = Z x √ g (cid:26) − χ R + v ( χ ) χ + 12 (cid:0) B ( χ ) − (cid:1) ∂ µ χ∂ µ χ (cid:27) . (151)The two functions B and v contain redundant information,since they can be changed by appropriate field transforma-tions.Using appropriate field transformations we can bring alarge class of effective actions with up to two derivatives into the generic form (151). For any positive and mono-tonically increasing function F ( χ ′ ) multiplying the gravi-ton kinetic term − R we can choose a normalization of thescalar field F = χ in order to bring the system to the Jor-dan frame. We can then use the residual transformationwithin the Jordan frame in order to obtain v = µ /χ suchthat B ( χ ) remains the only free function. Alternatively,we can obtain a constant scalar kinetic term at the prizeof a more complicated function v .Consider the transformation χ = h ( ˜ χ ) , g µν = ˜ χ h ( ˜ χ ) ˜ g µν . (152)This transforms the effective action (151) toΓ = Z x p ˜ g (cid:26) −
12 ˜ χ ˜ R + ˜ v ( ˜ χ ) ˜ χ + 12 (cid:0) ˜ B ( ˜ χ ) − (cid:1) ∂ µ ˜ χ∂ µ ˜ χ (cid:27) , (153)leaving the coefficient of the curvature scalar form-invariant. In terms of the variables ˜ χ and ˜ g µν the newfunctions ˜ B and ˜ χ read˜ B = B (cid:0) h ( ˜ χ ) (cid:1) (cid:18) ∂ ln h∂ ln ˜ χ (cid:19) , ˜ v = v (cid:0) h ( ˜ χ ) (cid:1) . (154)A pair of functions ( ˜ B, ˜ v ) describes the same model as thepair ( B, v ) if the two are related by eq. (154) with a suit-able choice of h ( ˜ χ ). The corresponding effective actions arerelated by a field transformation. A given model can be ex-pressed by a whole family of Jordan frames, parametrizedby h ( ˜ χ ).Of course, this equivalence also requires appropriatetransformations in the particle physics sector. For exam-ple, preserving the canonical kinetic term for a fermion field˜ ψ requires ψ = (cid:18) h ˜ χ (cid:19) ˜ ψ. (155)This implies that fermion masses ∼ ˜ χ remain form-invariant under the rescaling, with the same dimensionlesscoupling f , f p ˜ g ˜ χ ¯˜ ψ ˜ ψ = f √ gχ ¯ ψψ. (156)The rescaling leaves the dimensionless ratio betweenfermion mass and Planck mass, m ( ˜ χ ) / ˜ χ = f , invariant.
2. Kinetial crossover
We can employ the field transformations (152) in orderto bring a large class of potentials ˜ V ( ˜ χ ) to the “freeze form” V = µ χ . Indeed, any function ˜ v ( ˜ χ ) which decreasesmonotonically with limits ˜ v ( ˜ χ → → ∞ , ˜ v ( ˜ χ → ∞ ) → v = µ /χ by choosing h ( ˜ χ ) = µ p ˜ v ( ˜ χ ) . (157)0The choice of the effective action (2) is therefore rathergeneric, since a large family of potentials can be broughtto the particular form V = µ χ .A first example takes a constant potential˜ V = ¯ λ c , ˜ v = ¯ λ c ˜ χ , h ( ˜ χ ) = µ ˜ χ p ¯ λ c . (158)Eq. (154) yields B = ˜ B . (159)This relates the large- χ -behavior of the two models (A)and (B) in ref. [4].As a second example we may consider the models of ref.[68] ( A, α =const.),˜ v = µ ˜ χ − A m − A + ˜ χ − A , ˜ B = 4 α . (160)With h = ( m − A + ˜ χ − A ) ˜ χ A = χ (161)one finds B = ˜ B (cid:18) − A ) m − A Am − A + 2 ˜ χ − A (cid:19) , (162)where ˜ χ is related to χ by eq. (161). For large χ/m onehas χ = ˜ χ, B = ˜ B , while for χ → χ = m − A ˜ χ A , B = 4 ˜ B ˜ A = 4˜ α . (163)For these models the flow equation for B exhibits two fixedpoints with finite values of B , e.g. 4 / ˜ α for µ → ∞ and4 /α for µ →
0. This is the type of models investigated inrefs. [4, 46]. For µ → ∞ one has σ = − ∂ ln B/∂ ln χ → χ R equals ˜ χ up to a constantfactor, and V /χ R diverges for χ R →
0. This is not a fixedpoint in the sense of our previous discussion, but it couldrepresent a possible fixed point in terms of different vari-ables.As mentioned above, even much more general classes ofmodels can be described by a kinetial crossover. In theappendix E we discuss models where the coefficient of thecurvature scalar ˜ χ in eq. (153) is generalized to c ˜ χ + c µ , and map such models to the kinetial crossover form(2). This will shed light on the role of a possible term ∼ µ R for variable gravity models.
3. Potential crossover
Alternatively, we may use the transformation (154) in or-der to transform our models (2), (21) of a kinetial crossoverto an equivalent model with a potential crossover. For thispurpose we want to achieve a constant ˜ B , using for h asolution of the differential equation ∂ ln h∂ ln ˜ χ = s ˜ BB ( h ) . (164) Once h ( ˜ χ ) = χ is computed in this way we can computethe associated scalar potential V ( ˜ χ ) = ˜ χ v (cid:0) h − ( χ ) (cid:1) .As an example we consider the effective action (2) with B ( h ) obeying eq. (18),1 κB ( h ) − ln B ( h ) = ln (cid:18) hm (cid:19) . (165)For small h or large B one has the limiting behavior B − = hm (166)or ∂ ln h∂ ln ˜ χ = s ˜ Bhm . (167)The solution of eq. (167) involves an integration constant c h h = m (cid:18) c h − p ˜ B ln (cid:18) ˜ χm (cid:19)(cid:19) − . (168)This yields the potential˜ V = ˜ v ˜ χ = µ ˜ χ h = µ m (cid:20) c h ˜ χ − p ˜ B ˜ χ ln (cid:18) ˜ χm (cid:19)(cid:21) , (169)which vanishes for ˜ χ → V ( ˜ χ →
0) = µ ˜ B m (cid:20) ˜ χ ln (cid:18) m ˜ χ (cid:19)(cid:21) → . (170)On the other hand, for large h and small B we use B − = κ ln (cid:0) hm (cid:1) and therefore ∂ ln h∂ ln ˜ χ = s κ ˜ B ln (cid:18) hm (cid:19) . (171)The solution h = m exp ( κ ˜ B (cid:20) ln (cid:18) ˜ χm (cid:19) + ˜ c h (cid:21) ) (172)increases faster than ˜ χ for ˜ χ → ∞ . The correspondingpotential reads˜ V = µ m ˜ χ exp ( − κ ˜ B (cid:20) ln ˜ χm + ˜ c h (cid:21) ) . (173)The full potential makes a crossover from eq. (169) for˜ χ ≪ m to eq. (173) for ˜ χ ≫ m . The two integrationconstants c h and ˜ c h are related in order to ensure a smoothmatching, e.g. c h ≈ exp {− κ ˜ B ˜ c h / } .One may also choose a hybrid setting with a constantkinetic term ˜ B for χ →
0, while for χ → ∞ one keepsthe freeze frame V = µ ˜ χ . This is achieved by choosing h ( ˜ χ → ∞ ) = ˜ χ , while h ( ˜ χ →
0) is given by eq. (168).1
4. Primordial flat frame
Let us consider the frame where for χ → v and ˜ B are related by ∂ ln ˜ v∂ ln ˜ χ = − ˜ B ( ˜ χ ) + ∂ ln ˜ B∂ ln ˜ χ . (174)This is the condition for finding for the infinite past flatspace as a solution of the field equations derived from theaction (153) [4]. We can transform our crossover model (2),(5) to this “primordial flat frame” by a suitable choice of h in eq. (152). The function h ( ˜ χ ) has to obey a differentialequation which follows from˜ v = µ h , ˜ B = B ( h ) (cid:18) ∂ ln h∂ ln ˜ χ (cid:19) , (175)namely (cid:18) ∂ ln B∂ ln h (cid:19) ∂ ln h∂ ln ˜ χ − B (cid:18) ∂ ln h∂ ln ˜ χ (cid:19) +2 ∂∂ ln ˜ χ ln (cid:18) ∂ ln h∂ ln ˜ χ (cid:19) = 0 . (176)We are interested in h → B = ( m/h ) σ , ∂ ln B/∂ ln h = − σ . In this approximation eq. (176) isobeyed by ∂ ln h∂ ln ˜ χ = (2 + σ ) (cid:18) hm (cid:19) σ . (177)This yields the relation between χ and ˜ χ , χ = h ( ˜ χ ) = ˆ m (cid:18) ln ¯ m ˜ χ (cid:19) − σ , ˆ m = (cid:2) σ (2 + σ ) (cid:3) − σ m, (178)with ¯ m an integration constant. One infers˜ v = µ ˆ m (cid:18) ln (cid:18) ¯ m ˜ χ (cid:19)(cid:19) σ (179)and ˜ B = 2 + σσ ln (cid:16) ¯ m ˜ χ (cid:17) . (180)In the primordial flat frame ˜ B vanishes for ˜ χ → χ → B inthe frame of eqs. (2), (5). The dimensionless potential ˜ v =˜ V / ˜ χ diverges with an inverse power of a logarithm insteadof v ∼ χ − . Different frames can describe the same physicalsituation with rather different pictures. This extends to theform of the flow equations. For ˜ χ → ∂ ˜ B∂ ln ˜ χ = σ σ ˜ B ,∂ ˜ v∂ ln ˜ χ = −
22 + σ ˜ B ˜ v. (181) This transfers to the µ -flow equation for fixed ˜ χ and ˜ g µν (instead of fixed χ and g µν ) µ∂ µ ˜ B = − σ σ ˜ B , µ∂ µ ˜ v = 22 + σ ˜ B ˜ v. (182)On the level of a given quantum effective action the flowequations in different frames can be obtained from eachother by a simple transformation of field variables. Thequantum computation of the flow equations needs morecare. One may employ a field transformation which leavesthe functional integral invariant. It will involve, however,a Jacobian from the functional measure. In practice, acomputation is often done with the implicit assumption ofa unit Jacobian. This singles out a particular frame. Twosettings for which the “classical action” is related by a fieldtransformation, while both use the same definition of themeasure for the respective fields (e.g. unit Jacobian), resultin different models that do not yield equivalent predictionsfor observations. In other words, it is sufficient for the UV-fixed point of our model that there exists a frame for whicha quantum computation with unit Jacobian yields the flowequation (182) or, equivalently, eq. (22) supplemented with partialv/∂ ln µ = 2 v .
5. Asymptotic solution in the primordial flat frame
We could have started our discussion of inflation withthe quantum effective action (153),Γ = Z x √ g (cid:26) − χ R + ¯ λχ ln (cid:18) ¯ mχ (cid:19) + (cid:20) ln − (cid:18) ¯ mχ (cid:19) − (cid:21) ∂ µ χ∂ µ χ (cid:27) , (183)where we take σ = 2 for simplicity, ¯ λ = µ / ˆ m , and weomit the tilde on the fields. In the absence of radiationand matter the field equations (C.5), (C.6) read (cid:16) ¯ mχ (cid:17) − (cid:18) ¨ χχ + 3 H ˙ χχ (cid:19) + ln − (cid:18) ¯ mχ (cid:19) (cid:18) ˙ χχ (cid:19) +¯ λχ (cid:18) (cid:18) ¯ mχ (cid:19) − (cid:19) = 12 H + 6 ˙ H, (184)and (cid:18) H + ˙ χχ (cid:19) = 13 ln (cid:16) ¯ mχ (cid:17) (cid:18) ˙ χχ (cid:19) + ¯ λ χ ln (cid:18) ¯ mχ (cid:19) . (185)In leading order the solution for χ → , t → −∞ corre-sponds to flat space with slowly increasing χ according to H = 0 , ˙ χ = r ¯ λ χ ln (cid:18) ¯ mχ (cid:19) , (186)such that the sum of scalar potential and (negative) kineticterm vanishes. The approximate time evolution of χ isgiven implicitly by χ ln (cid:18) ¯ mχ (cid:19) = r λ ( t c − t ) − , (187)2with χ → t → −∞ .In the next to leading order the solution becomes H = c H ˙ χ ln (cid:16) ¯ mχ (cid:17) χ , δχ = c χ χ ln (cid:16) ¯ mχ (cid:17) , (188)with χ = χ + δχ and χ the leading order solution accord-ing to eq. (186). The field equations (184), (185) are bothobeyed for c H − c χ = 16 , (189)such that we are left at this stage with two free integrationconstants t c and c H . For c H = 0 the qualitative evolutionof the Hubble parameter reads H ≈ c H ( t c − t ) ln (cid:2) ˜ m ( t c − t ) (cid:3) , (190)with ˜ m varying only slowly with time. Geometry ap-proaches flat space in the infinite past ( t → −∞ ), with aslowly vanishing or diverging Robertson-Walker scale fac-tor depending on the sign of c H , a = c a ln − c H (cid:18) ¯ mχ (cid:19) . (191)For the particular solution c H = 0 , c χ = − / H = ˜ c H ˙ χ ln (cid:16) ¯ mχ (cid:17) χ . (192)In this case geometry approaches in the infinite pastMinkowski space, with a constant scale factor a ∞ accordingto a = a ∞ exp − ˜ c H ln (cid:16) ¯ mχ (cid:17) . (193)
6. Eternal universe
The geometry given by the solution (190) or (192), with χ according to eq. (186), is free of any singularity as longas χ remains finite. (The formal singularity of eq. (187)for t → t c is an artefact of the approximation, which isno longer valid for t near t c .) Space-time is geodesicallycomplete. In this frame it is straightforward to see thata universe given by our solution is eternal. It has existedsince the infinite past. As discussed in detail in ref. [68],physical time can be measured by the number of oscilla-tions of wave functions. Physical time indeed goes to mi-nus infinity in the limit t → −∞ . Discrete oscillation num-bers are the same in all frames, such that physical time isframe-independent. The eternity of the universe is there-fore independent of the chosen frame.Unphysical singularities in the Einstein frame arise froma singularity of the field transformation. (See ref. [68]for a more detailed discussion.) Indeed, a Weyl scaling g ′ µν = ( χ /M ) g µν brings the effective action (183) to theform Γ = Z x p g ′ n − M R ′ + ¯ λM ln (cid:18) ¯ mχ (cid:19) + M χ ln (cid:16) ¯ mχ (cid:17) ∂ µ χ∂ µ χ o . (194)This field transformation becomes singular for χ → λ ln (cid:18) ¯ mχ (cid:19) = exp (cid:16) − αϕM (cid:17) (195)we recover the form of eq. (34), with kinetial k = 2 α ¯ λ exp (cid:16) − αϕM (cid:17) . (196)For σ = 2 and ¯ λ = 8 µ /m this agrees with eq. (37),providing for a direct link to the discussion of inflation insect. III.
7. Free scalar field coupled to gravity
Another interesting frame change transforms the effec-tive action (2) to a scalar field theory without self interac-tions,Γ = Z x p ˜ g ( − f ( ˜ χ ) ˜ χ ˜ R + µ ˜ χ + ˜ K ∂ µ ˜ χ∂ µ ˜ χ ) , (197)with ˜ χ -independent ˜ K . This is achieved by transforma-tions that leave √ gχ invariant, g µν = 1 f ˜ g µν , χ = f ˜ χ. (198)The transformed kinetic coefficient becomes˜ K = f ( B − B − ∂ ln f∂ ln ˜ χ + (cid:18) B − (cid:19) (cid:18) ∂ ln f∂ ln ˜ χ (cid:19) ) . (199)For example, one may obtain ˜ K = 0 by solving for a given B ( f ˜ χ ) the differential equation for fB (1 + y ) = 6 (cid:16) y (cid:17) , y = ∂ ln f∂ ln ˜ χ . (200)In this frame the scalar field has only gravitational inter-actions. The cosmological field equation expresses ˜ χ as afunction of ˜ R by the implicit equation f + 12 ∂f∂ ln ˜ χ = 2 µ ˜ R . (201)The transformation (198) can be used in both ways. Aneffective action with constant ˜ K and non-trivial f ( ˜ χ ) canbe mapped to the form (2), with B = (cid:18) ∂ ln f∂ ln ˜ χ (cid:19) − " ˜ Kf + 6 (cid:18) ∂ ln f∂ ln ˜ χ (cid:19) . (202)3In particular, for f = ( ˜ χ/ ˜ m ) ˜ σ one finds B = (1 + ˜ σ ) − " ˜ K (cid:18) ˜ m ˜ χ (cid:19) ˜ σ + 6 (cid:18) σ (cid:19) = (cid:18) mχ (cid:19) σ + 32 ( σ − , (203)with σ = ˜ σ σ , m = (cid:2) ( σ − ˜ K ] σ ˜ m. (204)For σ > B = ( m/χ ) σ is thereforeequivalent to a positive constant ˜ K > f diverging ∼ χ ˜ σ , ˜ σ = − σ/ ( σ − σ = −
2, corresponds to σ = 2.On the other hand, the behavior (88) near the futurefixed point for χ → ∞ can be cast into the form (197) for˜ K = − , f = 1 + 16 κ ln ˜ χm . (205)The fixed point corresponds to f = 1, with flow equation ∂ t ( f −
1) = 6 κ ( f − . (206)There are two lessons to be learned from the discussionof this section. The first concerns the generality of ourdescription of the crossover by a varying kinetic term. Thesecond concerns the form of the µ -flow equation underlyingour approach. It depends on the choice of fields that arekept fixed as µ is varied, compare eqs. (182) with eq. (22)supplemented with µ∂ µ ln v = 2. The form of the flowequation depends o the frame. It transforms according toa variable change in a differential equation. VII. Conclusions
We have investigated the cosmological consequences ofa particle physics scenario for quantum gravity with anultraviolet (UV) and infrared (IR) fixed point. The exis-tence of an UV-fixed point renders quantum gravity non-perturbatively renormalizable (asymptotic safety). At thisfixed point the exact scale symmetry is not spontaneouslybroken, such that all particles are massless. It seems pos-sible that appropriate renormalized couplings obey asymp-totic freedom. Their running is governed, however, by large(non-perturbative) anomalous dimensions. These largeanomalous dimensions provide for a simple inflationary sce-nario. In the particle physics sector they could lead to apossible explanation of the gauge hierarchy for the elec-troweak symmetry breaking.For the IR-fixed point the exact scale symmetry is spon-taneously broken, resulting in massive particles and a mass-less dilaton. The ratio between the effective scalar poten-tial and the fourth power of the variable Planck mass van-ishes at this fixed point. As the fixed point is approached this ratio decreases to tiny values. In the Einstein framethis leads to an asymptotically vanishing effective cosmo-logical constant. Close to the fixed point the dilaton ap-pears as a pseudo-Goldstone boson withe a very small mass- the cosmon. The potential energy of the cosmon field isresponsible for dynamical energy.Dimensionless couplings can depend only on dimension-less ratios of quantities with dimension mass. In our casethis is χ/µ , where χ is the value of a scalar singlet field(cosmon) and µ the intrinsic mass scale appearing in theflow equations for the running couplings. The UV-fixedpoint is reached for µ → ∞ or χ →
0, while the IR fixedpoint corresponds to µ → , χ → ∞ . Cosmology describesa crossover from the UV-fixed point in the infinite pastto the IR-fixed point in the infinite future. This is real-ized by a cosmological solution with χ ( t → −∞ ) → χ ( t → ∞ ) → ∞ .The crossover between the two asymptotic fixed points isresponsible for the different epochs in cosmology. We pur-sue models for which the crossover occurs in two distantsteps, separated by a range of scales for which the flow ofcouplings is very slow. This range can be associated to theflow in the vicinity of an (approximate) “standard modelfixed point” (SM), see Fig. 1. The range of χ and associ-ated range in cosmological time where the SM-fixed pointdominates describes the radiation and matter dominatedepochs in cosmology. The UV-fixed point is responsible forthe inflationary epoch, which ends at the first step of thecrossover (UV → SM). The IR-fixed point will correspondto an (unknown) future scaling solution. The second stepof the crossover (SM → IR) entails a transition period forthe present cosmology for which dynamical dark energy(quintessence) dominates.The cosmology of our model involves four dimensionlessparameters besides the masses and couplings of particles ofthe standard model and some dark matter candidate: σ : anomalous dimension of the scalar at (or close to) theUV fixed point. It determines the spectral index n and tensor ratio r of the primordial fluctuations, cf.eq. (84). mµ : scale of the first step of the crossover. It fixes theamplitude A of the primordial fluctuations, see. eq.(56). κ : coefficient of the approach to the IR-fixed point. Itdetermines the fraction Ω lsh of early dark energy atlast scattering, eq. (124).˜ γ : present growth rate of the ratio neutrinomass/electron mass. In the Einstein frame itleads to a sizeable neutrino-cosmon coupling β = − ˜ γ p κ ln( M/m ). The combination ˜ γm ν determines the present fraction of dark energy Ω h (often called Ω Λ ), eq. (99).In addition, the present average neutrino mass m ν isnot yet experimentally determined - only lower and up-per bounds are established. We thus end with five un-known quantities that can be measured by cosmological4tests. At present the model seems comparable with ob-servations, with parameters σ ≈ . , ln( m/µ ) ≈ , κ ≈ . , ˜ γm ν ≈ . . Our model has the same number of free parameters asthe ΛCDM model (e.g. n, r, A , Ω h , m ν ). A quantum grav-ity computation could aim for a determination of σ and κ which may not depend strongly on the particle physicscontent of the model.The overall description of cosmology by our model is sim-ple. It describes all cosmological epochs by the dynamicsof a single scalar field, the cosmon. In the near future ourmodel is subject to interesting tests: the details of inflation(relation between n and r ), early dark energy and possi-ble consequences of large non-linear neutrino lumps. It isfascinating that a basic hypothesis about quantum gravityand the origin of mass, namely the existence of two fixedpoints and the necessary crossover between them, becomestestable by cosmology. Appendix A: Flow equations for effectiveaction
A central ingredient for this paper are the dimensionlessfunctions B ( χ/µ ) , C ( χ/µ ) , D ( χ/µ ) , E ( χ/µ ) and similarfunctions in the matter sector of the effective action. Wewill present here no computation of these quantities. Themain line of this paper makes an ansatz and explores itscosmological consequences. Nevertheless, in view of a fu-ture quantum gravity computation of functions as B ( χ/µ ),a few words are in order how their dependence on the ratio χ/µ arises. This is the purpose of this appendix.The status of B ( χ/µ ) is a renormalized function that ap-pears in the quantum effective action for which all quan-tum fluctuations have been included. It contains the infor-mation on one-particle irreducible vertices that obtain byfunctional derivatives of the corresponding scalar kineticterm in the effective action (2). These vertices are typi-cally evaluated for external momenta given by the intrinsicmass scale µ . (Sometimes momenta may be much smallerthan µ and can effectively be set to zero.) One could alsoview B as a µ -dependent dimensionless coupling. How-ever, the fact that B can only depend on χ/µ implies thatany µ -dependence translates to a field-dependent function B ( χ/µ ).
1. Relative mass scales
The key feature is the presence of two sets of scales in thequantum effective action, and therefore in the renormal-ized vertices. While µ denotes the set of all intrinsic massscales, the field χ accounts for the mass scales associatedto the spontaneous breaking of scale symmetry. In a cer-tain sense B ( χ/µ ) is an analogue to the universal Widomscaling function of magnetic systems, with µ standing forthe explicit scale symmetry breaking away from the criticaltemperature related to T − T c , while χ is the magnetiza-tion whose value can be dialed by a magnetic field. Anotheranalogue are renormalized dimensionless couplings in the standard model of particle physics. They depend on ex-ternal external momenta ∼ µ and particle masses whichare proportional to the value of the Higgs field ˜ h ∼ χ . (Inour setting we may use ˜ h = √ ǫ h χ , with ǫ h a constant for µ ≪ χ , cf. sect. V.) The flow equation for B ( χ/µ ) re-sults from the relative shift of the system of intrinsic massscales ∼ µ as compared to the system of “spontaneous”mass scales ∼ χ . For any practical computation one has tospecify these two systems.In our setting of variable gravity the spontaneous scale χ denotes the variable Planck mass. It acts as an effec-tive cutoff for the contribution of graviton fluctuations tovertices with external momenta Q ≪ χ . While gravi-tons remain massless for arbitrary χ , the contributions ofgraviton loops involve inverse powers of χ since they areproportional to the gravitational coupling ∼ χ − (vary-ing Newton’s “constant”). This leads to effective decou-pling, with contributions to the flow suppressed by powersof Q /χ . In a somewhat different context this has beenfound by the explicit computation in ref. [14]. An im-portant exception for the effective decoupling of gravitonfluctuations for Q ≪ χ concerns situations where rele-vant graviton propagators in loops are close to poles.For loop momenta q ≫ χ the higher order curva-ture terms ∼ C, D dominate the inverse propagators inthe gravity sector, whose general form reads symbolically(omitting constants etc.) Cq + χ q . Variation of χ canbe seen as the variation of the transition scale from effec-tive fourth-derivative gravity to effective second-derivativegravity. Furthermore, particle masses are proportional to χ . For example, the electron mass is given by m e = h e ˜ h = h e √ ǫ h χ = f e χ , with h e the Yukawa coupling of the elec-tron to the Higgs doublet. Variation of the spontaneousscale χ therefore corresponds to a simultaneous change ofthe Planck mass and the particle masses.In the freeze frame the intrinsic scale µ enters directlythe mass term for the cosmon. It induces an effective in-frared cutoff for the cosmon fluctuations, as given by thesquared renormalized mass µ R ∼ µ /B . (The cosmon massis not proportional to χ , in contrast to the other particlemasses.) Furthermore, we evaluate all couplings at exter-nal momenta Q ∼ µ . This choice is motivated by ourfinding that the cosmological solutions of the field equa-tions derived from the effective action (2), (3) are charac-terized by a Hubble parameter H which is proportional to µ . Fluctuations with wavelength larger than H − ∼ µ − do not contribute to the effective action for the cosmolog-ically relevant values and time derivatives of metric andscalar fields. This absence of long-wavelength fluctuationsis mimicked by external momenta Q ∼ H ∼ µ . A varia-tion of the intrinsic scale µ therefore reflects a variation ofexternal momenta combined with a variation of the scalarmass term. We should mention at this point that “naive”quantization depends on the frame. In a different framethe role of external momenta can look rather different, aswe will discuss below.Having determined the two sets of intrinsic and spon-taneous scales we can now establish the origin of the flowequations for functions as B ( χ/µ ). We may either keep5the particle masses and the Planck mass fixed (constant χ )and vary external momenta and scalar mass term simulta-neously (varying µ ). Equivalently, we may keep externalmomenta and scalar mass term fixed (constant µ ), and varysimultaneously the effective cutoff for particle and gravitonfluctuations (varying χ ).In our approach we keep only two sets of scales, with allparameters of dimension mass either proportional to µ orproportional to χ . This implies the simple relation betweenthe µ -flow at fixed χ and the χ -flow at fixed µ , µ ∂B∂µ | χ + χ ∂B∂χ | µ = 0 . (A.1)One could consider more generalized settings involvingmore than two sets of mass scales. For example, onecould investigate models with an explicit ultraviolet cut-off Λ, such that B depends on µ/ Λ besides χ/µ . In oursetting for renormalized B the cutoff Λ is considered asan intrinsic scale, with µ/ Λ fixed. In the presence of anultraviolet fixed point one can take the limit µ/ Λ → ∼ µ . Certain quantities may involve vastly differentexternal momentum scales.
2. Functional flow equation for effective averageaction
The functional flow equation for the effective averageaction [16] of dilaton quantum gravity [14] also involves twosets of mass scales: the infrared cutoff scale k and the scale χ set by the value of the scalar field. Coupling functionsas B depend now also on the ratio χ/k . Qualitatively wemay associate k with µ , since both scales act effectively asan infrared cutoff - one explicitly and the other indirectlyby virtue of non-zero external momenta.More precisely, the effective average action for our set-ting will involve the three sets of scales χ, µ and k . Oftenonly the highest effective infrared cutoff matters. Thus theflow of B (cid:16) χµ , χk (cid:17) is roughly independent of µ for k ≫ µ ,and independent of k for k ≪ µ . The quantum effectiveaction corresponds to k →
0. On the other hand, the pres-ence of an UV-fixed point implies the existence of a scalingsolution for Γ k . The scaling form of the effective aver-age action is independent of any intrinsic scale µ . Setting µ = 0 the scaling function B ∗ ( χ/k ) only depends on theratio χ/k . By virtue of the above decoupling properties wemay roughly identify B (cid:18) χµ ; k = 0 (cid:19) ≈ B ∗ (cid:16) χk , k = µ (cid:17) . (A.2)This connects the coupling function B ( χ/µ ) in the fullquantum effective action ( k →
0) to the scaling function ofthe effective average action B ∗ ( χ/k ).The identification (A.2) is only approximate for severalreasons. First, there are proportionality constants of orderone, replacing k = µ by k = c i µ on the r.h.s. of eq. (A.2),with coefficients c i depending on particular loop contribu-tions. They reflect the particular choice of cutoff as wellas the “final running” in the region k . µ before the flow stops. Second, the µ -flow equation needs to incorporateproperly the simultaneous change of the scalar mass pa-rameter which accompanies the change of scale of externalmomenta. Third, the form of the effective action (2) as-sumes a particular “canonical” choice of fields. The flow ofthe effective average action will typically not remain of thecanonical freeze form (2). One therefore needs to perform k -dependent field redefinitions [85, 86] in order to bringthe effective average action to the canonical freeze form atevery scale k .Despite these shortcomings several important generalfeatures can be inferred from the association between di-mensionless functions in the quantum effective action andscaling functions in the effective average action. An ultra-violet fixed point in the functional renormalization flow ofthe effective average action corresponds to k -independentrenormalized dimensionless couplings in the limit k → ∞ .Such an UV-fixed point in the k -flow is typically reflectedin an UV-fixed point for the µ -flow for the quantum ef-fective action. In the presence of an UV-fixed point thescaling function B ∗ ( χ/k ) is universal up to the dependenceon “renormalizable” couplings. These free renormalizablecouplings correspond to the relevant parameters for smalldeviations from the UV-fixed point. On the level of the µ -flow equations for the quantum effective action these freeparameters appear as integration constants for the solutionof the flow equation.
3. Scaling solutions and relevant parameters
A simple example for the connection between relevantparameters and free integration constants in the scalingsolution is the running of a non-abelian gauge coupling g in the presence of spontaneous symmetry breaking whichgives the gauge bosons a mass χ . The qualitative form ofthe flow equation ( c > k∂ k g | χ = − cg θ (cid:18) − χ k (cid:19) , (A.3)accounts for asymptotic freedom (UV-fixed point at g = 0)and the stop of the flow once k becomes smaller than thegauge boson masses. Here θ ( x ) is the heavy side function -smooth “threshold functions” approaching one for χ ≪ k and zero for χ ≫ k would lead to qualitatively similarresults. The scaling solution of eq. (A.3), g − = g − ∗ ( y ) = g − − c y θ (1 − y ) , y = χ k , (A.4)contains the free integration constant g . This correspondsto the value of the gauge coupling for k ≤ χ . One easilychecks that the scaling function g ∗ ( y ) is a fixed point of theflow equation for a fixed dimensionless ratio yk∂ k g | y = k∂ k g | χ + 2 y∂ y g = 0 . (A.5)According to eq. (A.2) the scaling solution g ∗ ( y ) can betaken over to the quantum effective action by the identifi-cation y = χ /µ .The IR-fixed point for the scaling solution (A.4) is rathertrivial. For any finite positive g the running of the cou-pling simply stops due to decoupling for k < χ . The6realization of an IR-fixed point by an effective stop of theflow due to excitations becoming heavy or interactions go-ing to zero is a rather generic phenomenon. It is, however,not the only way how an IR-fixed point can be realized.The scaling solution (A.4) is not the most general solu-tion of the flow equation (A.3). For the general solutionwe can replace the constant g by an arbitrary function of χ , g ( χ ) = g ( yk ) . (A.6)This is no longer a scaling solution due to the explicit de-pendence on k for fixed y . This dependence implies thepresence of a further scale as the ultraviolet cutoff Λ, suchthat g depends on the dimensionless ratios y and k/ Λ. Inthe limit k → y the general solution approachesa scaling solution, with constant g = g (0). This holdsprovided that the limit g ( χ / Λ →
0) is finite.We emphasize that a scaling solution is approached uni-versally in the infrared limit k → k → y and Λ corresponds to the limit Λ → ∞ atfixed y and k . In the presence on an ultraviolet fixed pointthe limit Λ → ∞ can be taken, removing any explicit de-pendence on Λ for renormalized dimensionless quantities.As a consequence g can only depend on y in this limit andthe scaling function has to be approached. There remains,however, some memory of the behavior close to the UV-fixed point. This corresponds to the relevant parametersor renormalizable couplings. The corresponding informa-tion has to appear in the form of free parameters for thescaling solution.The characteristic features discussed here are not specialfor the case of asymptotic freedom. As an example of adimensionless coupling λ with UV- and IR-fixed point weconsider the flow equation k∂ k λ | χ = c ( λ − λ IR )( λ − λ UV ) θ ( (cid:18) − χ k (cid:19) ,λ IR > λ UV , c > . (A.7)The scaling solution reads for y ≤ λ ∗ ( y ) = λ UV λ IR (1 − z ) + λ ( λ IR z − λ UV ) λ IR − λ UV z − λ (1 − z ) ,z = y c ( λ IR − λ UV ) , λ UV ≤ λ ≤ λ IR , (A.8)and λ = λ for y ≥
1. In the UV-limit y → λ ∗ ( y ) approaches λ UV , while the IR-limit is givenby λ ∗ ( y → ∞ ) = λ . The free integration constant λ de-notes the value of the coupling when the crossover between λ UV and λ IR is stopped by the decoupling for k < χ . For λ very close to λ IR the value of λ ∗ ( y ) remains close to thisvalue for a large range of y . The discussion of the generalsolution is similar to our first example, with λ dependingon the combination yk / Λ = χ / Λ .
4. Stages of the flow in the freeze frame
The analogy between the µ -flow of the quantum effectiveaction and the scaling solution of the effective average ac-tion helps to visualize which type of fluctuations contributeto quantities as B ( y ) = B ( χ /µ ) for different ranges of y .For y . y →
0. (The sym-bols . , & denote here order of magnitude estimates.) For y & χ . Thoseinclude the cosmon. More precisely, the propagators forthe scalar field χ and the scalar degrees of freedom in themetric mix and the cosmon is associated with a suitableeigenstate.Beyond the cosmon many particles of the standard modelhave masses much smaller than χ . In a grand unified theorythere would be particles with mass around (10 − − − ) χ that decouple once µ gets smaller than this value. Theparticles of the standard model have masses substantiallysmaller than χ due to the electroweak gauge hierarchy andthe small ratio between the QCD-scale and the Planckscale. They affect the running of the standard modelcouplings according to the well known perturbative β -functions. (The influence on the flow of B is not knownso far.) This standard model flow stops effectively once µ drops below the χ -dependent electron mass m e ( χ ) ≈ · − χ . Thus for 1 ≪ y . · one expects a range of“standard model flow”. In addition to the particles of thestandard model also the cosmon and possibly the gravitonand the scalar gravitational degree of freedom contributeto the flow. These contributions could actually dominatethe flow of B .The standard model flow ends at a value of y that cor-responds to a cosmological epoch before the electroweakphase transition. (Recall that the value of y relevant forpresent cosmology is around 10 .) In the Einstein framethe end of the standard model flow corresponds to a Hub-ble parameter of the order of the electron mass, cosmictime of the order m − e , or temperature in the range 10 GeV. For y & the running of the standard model cou-plings stops effectively and the flow is characterized by thestandard model fixed point.In the vicinity of the standard model fixed point onlyneutrinos, photons, cosmon and possibly gravitational de-grees of freedom as well as other light particles beyondthe standard model (e.g. very light scalar fields for darkmatter [87, 88]) contribute to the flow. Neutrinos will de-couple once µ drops below the neutrino mass. Accordingto our assumption the flow in this range is unstable in the“beyond standard model (BSM)-sector”, exhibiting a re-maining (marginally) relevant coupling. The second stepof the crossover occurs once this relevant deviation fromthe standard model fixed point becomes large. Neutrinomasses are directly sensitive to the BSM-sector, such thatthe second stage of the crossover becomes first “visible” inthese quantities.We end this short qualitative “history” of the flow withthe remark that the effect on the kinetial B may be indi-7rect. For example, a contribution to the flow of the po-tential (e.g. a running χ -independent term) translates toa flow of B by field transformations, as discussed in sect.VI and appendix E.
5. Flow equations in different frames
Let us assume for this paragraph that the leading termin the potential for large χ is not a term quadratic in χ asin eq. (2), but rather a constant V = ˜ µ . A cosmologicalmodel of this type has been discussed in ref. [4], model(B). Realistic cosmology requires ˜ µ ≈ · − eV. Thecharacteristic size of the Hubble parameter is H ≈ ˜ µ / ˜ χ ,with ˜ χ the scalar field in this frame. We can take overthe preceding discussion with the association µ = ˜ µ / ˜ χ or˜ y = ˜ χ / ˜ µ = √ y .Once the kinetial ˜ B (˜ y ) is found in this setting, it canbe transformed to the one in the freeze frame, B ( y ), bythe variable transformation discussed in sect. VI, e.g. eq.(159). This demonstrates a general feature. Typically, aquantum computation of the flow equation will be donein a particular frame, e.g. assuming implicitly a simplefunctional measure. The translation to the freeze framecan then be done on the level of the quantum effectiveaction. Appendix B: Fixed points and crossover fordimensionless couplings
In this appendix we discuss the general structure of thecrossover between two fixed points for a dimensionless cou-pling. For this purpose we use a simple example for theflow equations. We apply these findings to a discussion ofthe behavior of neutrino masses during the second stage ofthe crossover. For a dimensionless coupling h dependingon χ/µ we formulate the flow equation directly in terms ofthe χ -dependence for fixed µ .Consider a dimensionless coupling h whose dependenceon χ obeys the flow equation ∂h ∂ ln χ = c ( h − f )( h − f ) , (B.1)with f and f the values of two fixed points for h , with0 < f < f . We take c > f is approachedfor χ → f for χ → ∞ . The solution of eq. (B.1) isgiven implicitly by h − f f − h = (cid:18) χχ (cid:19) − c ( f − f ) . (B.2)As χ increases from zero to infinity this describes for h acrossover from the fixed point at f to the one at f . Thefree integration constant χ is proportional to µ . It de-termines at which value of χ/µ the crossover occurs. Thisdepends how “close” a given trajectory is to the UV-fixedpoint. The ratio χ /µ can therefore be exponentially largeor small. For sufficiently large χ one has approximately h = f + ( f − f ) (cid:18) χχ (cid:19) − c ( f − f ) . (B.3)Then the relative change of h ,1 h ∂h ∂ ln χ = − c ( f − f ) f (cid:18) χχ (cid:19) − c ( f − f ) , (B.4)becomes tiny forln( χ/χ ) ≫ (cid:2) c ( f − f ) (cid:3) − . (B.5)Particle masses can be written as m A = h A ( χ ) χ , with anappropriate dimensionless coupling function h A ( χ ). If h A obeys eqs. (B.1), (B.5) the particle mass m A scales (al-most) proportional to χ . This is what we will assume forall particles of the standard model except for neutrinos.The scaling m A ∼ χ indicates dilatation or scale symme-try, as appropriate for a fixed point.For these particles we associate χ with m . For nucle-osynthesis and the subsequent epochs of cosmology the ra-tio χ/m is already huge. For sufficiently large c ( f − f )the χ -dependence of couplings and mass ratios for stan-dard model particles will be too small to be observable. Forsmaller c ( f − f ), however, a tiny residual χ -dependenceof couplings could result in an observable time variation.The couplings during nucleosynthesis may then be slightlydifferent from the present ones. Eq. (B.4) relates the timevariation during nucleosynthesis with the one in the presentcosmological epoch.Let us next discuss the crossover from f to f , special-izing to f ≪ f . We may consider the crossover region f ≪ h ≪ f where we approximate h = f (cid:18) χχ (cid:19) − cf . (B.6)For the particular flow equation (B.1) this region is char-acterized by a constant anomalous dimension ∂ ln h ∂ ln χ = − cf . (B.7)We will assume that neutrino masses show this type ofcrossover behavior.Neutrino masses are characterized by the seesaw formula, m ν = h ν ϕ H M B − L , (B.8)with ϕ H the Fermi scale (expectation value of the Higgsdoublet) and M B − L a characteristic high mass scale where B − L symmetry is violated. Severe bounds on the timevariation of electron over proton mass indicate that ϕ H /χ must be close to a fixed point already for values of χ char-acteristic for nucleosynthesis. A first scenario may assume h = M B − L /χ , with h obeying the flow eq. (B.1). With ϕ H ∼ χ eqs. (B.7), (B.8) yield˜ γ = 12 ∂ ln( m ν /χ ) ∂ ln χ = cf . (B.9)8A non-trivial scaling m ν ∼ χ γ may correspond to sucha crossover situation.Other forms of a crossover, with ˜ γ depending on χ , areconceivable as well. With ˜ m = m ν /χ , any positive con-tinuous function ˜ γ ( ˜ m ) with two different zeros at ˜ m and˜ m describes a crossover between ˜ m = ˜ m for χ → m = ˜ m for χ → ∞ .For our second scenario we take˜ γ = ˜ m − ˜ m ˜ m ˜ m − ˜ m ˜ m , (B.10)with constant fixed point values ˜ m and ˜ m , ˜ m ≪ ˜ m .Again, a non-zero value of ˜ γ reflects the χ -dependence of M B − L /χ . For the crossover region ˜ m ≪ ˜ m ≪ ˜ m we canapproximate ˜ γ = ˜ m ˜ m . (B.11)The corresponding solution of eq. (B.9), ˜ γ = ∂ ln ˜ m/ (2 ∂ ln χ ), namely˜ m = m ν χ = ˜ m ln (cid:16) ¯ χ ν χ (cid:17) , (B.12)diverges for χ approaching the constant ¯ χ ν . This is, how-ever, outside the validity of the approximation. For ˜ m ap-proaching ˜ m eq. (B.10) implies that the increase with χ isstopped. In the crossover region, however, the fixed pointat ˜ m is not yet visible. We could also multiply the r.h.s.of eq. (B.10) with a constant. Within the crossover regionthis constant can be absorbed into a redefinition of ˜ m . Atthis point the χ -dependence of the average neutrino massinvolves two parameters, ˜ m and ¯ χ ν . We will see that itcorresponds to the setting of ref. [7].The parameter ˜ m is given by the ratio between the aver-age neutrino mass and the Planck mass in earlier epochs ofcosmology, before the crossover in the neutrino sector setsin. Taking in eq. (B.8) an “early value” M B − L /χ ≈ − ,as appropriate for B − L violation at some scale character-istic for grand unification, and ϕ H /χ ≈ − , as given bythe electroweak gauge hierarchy, we estimate˜ m = 10 − h ν , (B.13)with h ν typically smaller than one. (For these estimates weemploy the present value of χ , namely M = 2 . · eV.)For the present value of ˜ m one has˜ m ( t ) = (cid:18) m ν ( t )0 . (cid:19) · − . (B.14)This is well compatible with our assumption that for thepresent cosmological epoch the neutrino masses are in thecrossover region,˜ γ ( t ) = ˜ m ( t )˜ m − (cid:16) ¯ χ ν M (cid:17) = 10 h ν (cid:18) m ν ( t )0 . (cid:19) − ≫ . (B.15)For ˜ γ ( t ) = 9 the ratio M B − L /χ has decreased at presentby a factor of ten as compared to its early value. Appendix C: Field equations for variablegravity and asymptotic solutions
In this appendix we discuss field equations and solutionsof variable gravity. For the solutions we do not attempt togive a complete overview but rather concentrate on a fewcharacteristic ones.
1. Field equations
In this appendix we investigate the field equations de-rived from the effective actionΓ = Z x √ g (cid:26) − χ R − C R + B ( χ ) − ∂ µ χ∂ µ χ + V ( χ ) (cid:27) , (C.1)with constant C . We have omitted the term ∼ D in eq.(3) since it does not contribute to the field equations fora spatially flat Robertson-Walker metric if D is constant.The gravitational field equation is obtained by variation ofthe effective action (C.1) with respect to the metric, χ ( R µν − Rg µν ) + D χ g µν − D µ D ν χ +( B − (cid:18) ∂ ρ χ∂ ρ χg µν − ∂ µ χ∂ ν χ (cid:19) + V g µν (C.2)+ C (2 RR µν − R g µν + 2 D Rg µν − D µ D ν R ) = T µν . Here D µ denotes the covariant derivative, D = D µ D µ ,and T µν is the energy momentum tensor. The cosmon fieldequation is given by( B − D χ + 12 ∂B∂χ ∂ µ χ∂ µ χ = ∂V∂χ − χR − q χ , (C.3)with q χ the contribution from particles with χ -dependentmass. Contracting eq. (C.2) yields a differential equationfor the curvature scalar if C = 0,6 CD R − χ R + 6 χD χ + B∂ µ χ∂ µ χ + 4 V = T µµ . (C.4)For a Robertson-Walker metric with vanishing spatialcurvature a time dependent homogeneous scalar field obeys( B − χ + 3 H ˙ χ ) + 12 ∂B∂χ ˙ χ + ∂V∂χ − χ (12 H + 6 ˙ H ) = q χ , (C.5)while the (0 , T = ρ )3( χH + ˙ χ ) = B χ + V + 18 C ( ˙ H − H ¨ H − H ˙ H ) + ρ. (C.6)Taking a time-derivative of eq. (C.6), adding eq. (C.5)multiplied by ˙ χ and using eq. (C.4) yields the generalizedconservation equation ( T µµ = − ρ + 3 p )˙ ρ + 3 H ( ρ + p ) + q χ ˙ χ = 0 . (C.7)We will use eqs. (C.5), (C.6) and (C.7) as the three in-dependent equations which determine the time evolution.We observe that the contribution from the term ∼ CR vanishes for constant H .9
2. Asymptotic solutions
For primordial cosmology we neglect T µν and q χ . In theabsence of matter and radiation eqs. (C.5) and (C.6) con-stitute two non-linear second order differential equationsfor χ and H . For V = µ χ and B = ( m/χ ) σ they admitthe simple family of solutions χ = 0 , H = H , (C.8)with H an arbitrary constant. For positive H ˙ χ theseasymptotic solutions can be approached in the infinite past t → −∞ provided H > µ √ . (C.9)Small deviations from the asymptotic solution (C.8) growas time increases.For the case H = µ/ √ σH = µ √ , χ = m √ σµ ( t c − t ) ! σ . (C.10)For C = 0 and arbitrary H > µ/ √ H = H + δH,χ = m (cid:18) H σ (6 H − µ )( t c − t ) (cid:19) σ , (C.11)provided that δH vanishes for t → −∞ according to thesolution of the equation˙ H = χ C (cid:18) µ H − (cid:19) . (C.12)Eq. (C.12) is necessary for the approximate solution ofeq. (C.6). It is compatible with | δH ( t → −∞ ) | → σ <
2. Then typical solutions that can be extendedtowards the infinite past switch from an asymptotic so-lution (C.11) for t → −∞ towards the scaling solution(C.10) as time increases. We recall that such solutionswith H = µ / C = 0 , σ < ∼ CR we evaluate its contribution to equation (C.6).For t → −∞ we investigate solutions of the type H = bµ + f ( t c − t ) − η , χ = ¯ χ ( t c − t ) − ζ . (C.13)The term ∼ C in eq. (C.6) contributes in leadingorder − CH ˙ H = 108 Cηf b µ ( t c − t ) − (1+ η ) , to becompared with V = µ ¯ χ ( t c − t ) − ζ and ( B/
2) ˙ χ =( m σ / ζ ¯ χ − σ ( t c − t ) − ζ ( σ − . For 2 ζ < η the po-tential V dominates the r.h.s of eq. (C.6) for t → −∞ ,implying b = 1 / √
3. For the vicinity of the scaling solution(C.10) this turns out to be indeed the leading behavior for H for t → −∞ . On the other hand, for the vicinity of theasymptotic solutions (C.11) for H = µ/ √ ζ = 1 + η such that the term ∼ C is of equal importance as V on the r.h.s. of eq. (C.6). We will discuss the twocases separately.For H = µ/ √ χ = 2 √ µm − σ χ σ , (C.14)such that eq. (C.13) is obeyed with ζ = 1 σ , ¯ χ = m √ σµ ! σ . (C.15)This is the scaling solution (C.10). In the infinite past t → −∞ one has χ = 0 and realizes eq. (12).For the next to leading contribution to H one finds f ( t c − t ) − η = − σ ( t c − t ) − − f F c ( t c − t ) − − η + σ , (C.16)with F c = 6 √ Cηµm (cid:18) σµ √ (cid:19) σ . (C.17)For σ < t → −∞ , such that the term ∼ C dominates the next toleading correction to H . A solution for t → −∞ is thereforegiven by eq. (C.13) with η = 2 σ , f = − σF c . (C.18)As t increases towards t c the l.h.s of eq. (C.16) increasesfaster than the r.h.s. if σ <
2. It equals the term ∼ C at atransition time t tr given by t c − t tr = | F c | − σ − σ = √ σµ (cid:18) | C | µ m (cid:19) − σ − σ . (C.19)After the transition, for t c − t ≪ t c − t tr , the next to leadingorder contribution to H switches to η = 1 , f = − / (6 σ ) , (C.20)and the contribution of the higher curvature term ∼ C becomes negligible. We observe that for C of the orderone the dimensionless ratio µ ( t c − t tr ) is large due to thesmall ratio µ /m . If we associate t c roughly with the endof inflation this implies that the higher order curvatureterm ∼ C becomes negligible long before the observablefluctuations cross the horizon.We conclude that for σ < ∼ CR does not influence the leading behaviour of the scalingsolution for H and χ . Only the next to leading terms inthe solution for t → −∞ are influenced by C = 0. Thisinfluence ends effectively at t tr , long before the observableprimordial density fluctuations leave the horizon. For theobservable properties of inflation the role of the term ∼ CR is negligible.0For σ > δH occurs in the inverse order. For t → −∞ one findsthe solution (C.20), while for t → t c the term involving C becomes dominant in eq. (C.16). For t > t tr the solutionformally switches to eq. (C.18) and C seems to matter. For σ > t tr is very close to the endof inflation at t c , however. At this time the approximation(C.13) is no longer valid. For σ > ∼ C is negligible for all t , with δH = − σ ( t c − t ) − . (C.21)Finally, we may also consider the boundary case σ = 2for which all three terms in eq. (C.16) have the same timedependence, η = 1 , f = − F c + 1) , F c = 24 Cµ m . (C.22)The quantitative influence of the term ∼ C is negligible forall t due to the tiny ratio µ /m .We next turn to the asymptotic solutions (C.11) with H = µ/ √ b = 1 / √
3. For σ < ζ =1 /σ, η = 2 /σ − H reads H = H + f ( t c − t ) − − σσ . (C.23)The constant f is given by f = m ( µ − H ) σ CH (2 − σ ) (cid:18) H σ (6 H − µ ) (cid:19) σ . (C.24)This type of solution is only valid for σ < t → −∞ . For example, asolution with χ = 0 , ˙ χ = 0 can be found for2 H ¨ H = ˙ H − H ˙ H. (C.25)This differential equation admits solutions of the type(C.13) that approaches flat space in the infinite past ( b =0). With η = 1 , f = − the evolution of the scale factordescribes a shrinking universe H = −
12 ( t c − t ) − , a = a √ t c − t (C.26)In this case the past infinite universe is flat with infinitescale factor, similar to the future in the radiation domi-nated Friedman universe, a = a √ t − t c . (The time re-flected solution a = a √ t − t c also solves eq. (C.25).) Thistype of solution with negative H seems not to be connectedwith solutions that lead to realistic cosmologies for latertimes.
3. Renormalized scalar field
We are interested in the general behavior of solutionsclose to the UV-fixed point, e.g. for small χ . For χ close to zero it is advantageous to write the field equations interms of the renormalized scalar field χ R = 22 − σ √ Bχ , B = (cid:18) mχ (cid:19) σ . (C.27)With ˙ χ = B − / ˙ χ R , ¨ χ = B − (cid:18) ¨ χ R + σ − σ ˙ χ R χ R (cid:19) ,B ¨ χ + 12 ∂B∂χ ˙ χ = B / ¨ χ R , (C.28)and approximating B − ≈ B , the field equations (C.5),(C.6) read in the absence of matter and radiation¨ χ R + 3 H ˙ χ R = 2 − σ B (12 H + 6 ˙ H − µ ) χ R (C.29)and 36 CH ¨ H = 18 C ( ˙ H − H ˙ H )+ (cid:18) − σ (cid:19) B − ( µ − H ) χ R + 12 ˙ χ R − − σ ) B − H ˙ χ R χ R . (C.30)Once B − is expressed in terms of χ R , B − = (cid:18) (2 − σ ) χ R m (cid:19) σ − σ , (C.31)this form is suitable for numerical solutions. (At a pointwhere H vanishes (or for C = 0) the r.h.s. of eq. (C.30)has to be equal to zero.)We concentrate on σ < χ → χ R →
0. One recovers the simple solution χ R = 0 or χ = 0with an arbitrary constant value of the Hubble parameter H = H . The stability of this solution depends on thevalue of H . For H < µ/ √ χ R decreases towards zero. On the other hand, for H > µ/ √
6a perturbation in χ grows and the solution with χ = 0 isunstable. We are mainly interested in the second type ofsolution where χ moves away from the UV-fixed point at χ = 0 as time increases.It is useful to express the field equations in terms of thepotential for the renormalized field, V = µ m (cid:18) (2 − σ ) χ R m (cid:19) − σ . (C.32)The scalar field equation (C.29) reads¨ χ R + 3 H ˙ χ R = − ∂V∂χ R − H + 3 ˙ Hµ ! . (C.33)We observe that for H ≈ µ / χ R increases with in-creasing time.1Eq. (C.30) can be written in the form36 CH ¨ H = 18 C ( ˙ H − H ˙ H ) + V (cid:18) − H µ (cid:19) + 12 ˙ χ R − H ˙ χ R µ ∂V∂χ R . (C.34)We observe that eqs. (C.33) and (C.34) can be directly de-rived from the effective action (8), (9) for the renormalizedscalar field,Γ = Z x √ g (cid:26) ∂ µ χ R ∂ µ χ R + V ( χ R ) − V ( χ R )2 µ R − C R (cid:27) . (C.35)For a general effective action (2), (3) the relation between χ R and χ can be extended to arbitrary B in the form ∂χ R ∂χ = √ B − . (C.36)This coincides with eq. (C.27) if we replace B by B −
4. Numerical solutions
We have investigated numerically the coupled system ofdifferential equations (C.33), (C.34), (C.32) for constant σ <
2, starting at some time t with “initial conditions”for χ R , ˙ χ R , H and ˙ H . For positive H ( t ) we find that for alarge range of initial values for ˙ χ R and ˙ H both χ R and H approach very rapidly almost constant values. After thisfirst stage one observes a second stage of slow evolutionaccording to the approximate equations˙ χ R = − H ∂V∂χ R (cid:18) − H µ (cid:19) , ˙ H = V CH (cid:18) − H µ (cid:19) . (C.37)Part of the memory of the initial conditions has been lostat this stage. Finally, the evolution speeds up if V and ∂V /∂χ R become large enough, typically for χ R of the order2 m/ (2 − σ ). At this point B is of the order one, roughlycorresponding to the end of an inflationary epoch.This behavior is illustrated in Figs. 4 and 5 which showthe time evolution of χ R and H for two different initialvalues. Parameters are σ = 1 . , m/µ = 3 , C = 0 . µ . We observe an extremely slow evolution, recallingthat the time unit µ − amounts to 10 yr. For realisticlarger ratios of m/µ this would be even more extreme. Wehave compared our numerical results with solutions of theapproximate equations (C.37). The difference is not visiblein these plots. The green curve corresponds to the scalingsolution (C.10) with H = µ/ √
3, visible by the constantvalue of H . The other two solutions move towards thissolution.For an analytic discussion of eq. (C.37) we may intro-duce a new evolution variable z by dzdt = V, (C.38) FIG. 4: Slow time evolution of the renormalized scalar field χ R . For a large range of initial parameters the family of suchasymptotic solutions is approached very rapidly.FIG. 5: Slow time evolution of the Hubble parameter. such that the equations (C.37) read ∂χ R ∂z = − − σ ) (cid:18) H − Hµ (cid:19) ,∂H∂z = 1108 C (cid:18) H − µ (cid:19) . (C.39)The evolution of H has a fixed point at H = µ/ √ z . Close to the fixed point χ R increases monotonically with z , χ R = (cid:18) z √ − σ ) µ (cid:19) . (C.40)On the other hand, for H < µ/ √ χ R with increasing z .In particular, for C > H ≪ µ one finds approxi-mately H ( z ) = (cid:16) z C (cid:17) , (C.41)and χ R = χ R − − σ (36 C ) z . (C.42)Since χ R must be positive eq. (C.42) holds only for z
6. Fora very large class of initial conditions the general solutionapproaches very rapidly this family of attractor solutions.The attractor solutions can be extended to the infinite past,while this does not hold for neighboring solutions. Theattractor solutions obey the approximate field equations(C.37). The slow evolution according to these equationsentails an approach of all attractor solutions towards theparticular scaling solution (C.10). This slow evolution endsonce t is sufficiently close to t c such that B reaches a valuearound one for χ around m .We show the fast approach to the family of attractorsolutions in figs. 6 and 7 which compare the time evolutionof χ R and H for initial conditions given at some arbitrarytime t which the scaling solution (C.10). (Parameters areagain σ = 1 . , m/µ = 3 , C = 0 .
5. Crossing smoothly the big bang
For fig. 6 we have chosen initial conditions such that χ R ( t ) switches from negative to positive values. From eq.(C.27) we infer that also χ crosses zero at this moment( σ < χ corresponds to the big bangsingularity in the Einstein frame. We have therefore estab-lished an explicit solution of field equations which crossfrom a pre-big-bang regime to an after-big-bang regime.In the freeze frame this crossing is completely regular. Nosingularity appears in the field equations for a vanishingfield value of χ . The big bang singularity in the Einsteinframe is a pure artefact of the choice of fields which be-comes singular for χ = 0. In the freeze frame it takes only a finite interval in physi-cal time (as measured by the number of oscillations of wavefunctions) in order to cross from negative to positive valuesof χ . Since physical time does not depend on the frame,we conclude that the big bang singularity in the Einsteinframe is reached for a finite physical time for this type ofsolution. This contrasts to the asymptotic solutions wherethe big bang is reached only at an infinite interval of phys-ical time in the past. FIG. 6: Fast approach of the renormalized scalar field χ R tothe family of asymptotic solutions. In the Einstein frame thebig bang singularity is crossed when χ R goes through zero.FIG. 7: Fast approach of the Hubble parameter to the almostconstant value of the asymptotic solution. Appendix D: Two scalar field description
In this appendix we explain several features of the so-lutions discussed in appendix C in terms if an equivalentformulation with two scalar fields. We also discuss the pos-sibility of a periodic crossing of the “big bang singularity”for the initial cosmology of our model.
1. Formulation with two scalar fields
The higher curvature invariant ∼ C in eq. (C.1) entailsan additional physical scalar degree of freedom if C is apositive constant. (For the model (2), (3) we specialize toconstant C >
0, while we omit a constant D since it doesnot contribute to the field equations.) Some aspects of thebehavior of primordial cosmology become more apparentif we transform the effective action (C.1) to an equivalentmodel with two scalar fields.3For this purpose we add to Γ a term∆Γ = C Z x √ g ( φ − R ) , (D.1)such thatΓ ′ = Γ + ∆Γ = Z x √ g (cid:26) K ∂ µ χ∂ µ χ + µ χ −
12 ( χ + 2 Cφ ) R + C φ (cid:27) , (D.2)where K = B − . (D.3)Inserting the solution of the field equation for φ , namely φ = R , into Γ ′ we recover Γ. The field equations derivedfrom Γ and Γ ′ are therefore equivalent. The term ∼ CR is effectively replaced by the interactions of the new scalarfield φ . The field equations are now of second order.Employing a transformation of the metric, g µν = M χ + 2 Cφ g ′ µν = ¯ w g ′ µν , (D.4)one has R = ¯ w − (cid:8) R ′ − D ln ¯ w − ∂ µ ln ¯ w ∂ µ ln ¯ w (cid:9) , (D.5)where R ′ and the covariant derivatives on the r.h.s. areformed with the metric g ′ µν . In the new frame the effectiveaction readsΓ ′ = Z x p g ′ (cid:26) − M R ′ + V ′ + L ′ kin (cid:27) , (D.6)with V ′ = M ( χ + 2 Cφ ) (cid:18) µ χ + C φ (cid:19) , (D.7)and L ′ kin = M χ + 2 Cφ ) (cid:8)(cid:0) ( K + 6) χ + 2 Cφ (cid:1) ∂ µ χ∂ µ χ + 12 Cχ∂ µ χ∂ µ φ + 6 C ∂ µ φ∂ µ φ (cid:9) . (D.8)It describes standard Einstein gravity (with M the reducedPlanck mass) coupled to two scalar fields χ and φ .It is instructive to study the potential V ′ ( χ, φ ) as a func-tion of φ for fixed χ . For φ → ∞ it approaches the constant V ′∞ = lim φ →∞ V ′ ( χ, φ ) = M C , (D.9)while for φ → φ → V ′ ( χ, φ ) = M µ χ . (D.10)With ∂V ′ ∂φ = CM χ ( φ − µ )( χ + 2 Cφ ) (D.11) we observe a partial minimum for φ min = 4 µ . (D.12)We will see that this partial minimum corresponds to thescaling solution (C.10).For φ = 4 µ one has a valley in the potential V ′ ( χ, φ ).Along this valley the potential V ′ ( χ, φ min ) = µ M χ + 8 Cµ (D.13)vanishes for χ → ∞ . As time increases φ typically settlesat φ min , such that the kinetic term becomes L ′ kin [ χ, φ min ] = M (( K + 6) χ + 8 Cµ )2( χ + 8 Cµ ) ∂ µ χ∂ µ χ. (D.14)As a result of the decaying potential (D.13) the behaviourof the cosmological solution is then characterized by anincrease of χ to values χ ≫ µ . Asymptotically, for t →∞ , χ increases to infinity and the effective cosmologicalconstant V ′ vanishes.For small | Cφ/χ | ≪ , and Cµ /χ ≪
1, we can expand V ′ = M µ χ + CM χ ( φ − µ φ ) , (D.15)as well as L ′ kin = ( K + 6) M χ ∂ µ χ∂ µ χ (D.16)+ 3 C M χ ∂ µ φ∂ µ φ + 6 CM χ ∂ µ χ∂ µ φ. In the limit where derivatives of χ can be neglected φ de-scribes a stable scalar field. Depending on the relativesize of the kinetic term it will perform damped oscilla-tions around its minimum at φ = 4 µ and finally settlethere, or it will reach this minimum in an overdamped ap-proach. Realizing that φ equals the curvature scalar in thefreeze frame, R ≈ H , we recover indeed the leading be-haviour of the scaling solution in the freeze frame (C.10), H ≈ p R/
12 = p φ min /
12 = µ/ √
3. The time evolutionof χ induces additional terms in the field equation for φ which do not, however, affect the stability of the model.For χ ≫ √ Cµ we can to a good approximation neglect theinfluence of the term ∼ CR and set C = 0.We next turn to solutions in primordial cosmology thatfeature small values of χ . Expanding V ′ in powers of χ / CφV ′ = M C + M C φ (2 µ − φ ) χ + · · · , (D.17)we find a positive quadratic term ∼ χ for φ < µ , whileit turns negative for φ > µ . The field equations admitthe solution χ = 0 , φ = φ = const. , H = M C . (D.18)4Recalling the relation between φ and the curvature scalarin the freeze frame this corresponds to the family of asymp-totic solutions (C.8). For a stable kinetic term this solutionis stable for φ < µ and unstable for φ > µ . In ac-cordance with the findings of appendix C (cf. eq. (C.9)),the scalar field χ increases (for increasing time) for φ =12 H > µ , while it goes to zero for φ = 12 H < µ .We observe the difference of the Hubble parameter in thefreeze frame (denoted here by H ) from the one in the Ein-stein frame with two scalars, as in eq. (D.18). All valuesof H in the freeze frame correspond to the same H in eq.(D.18), while the corresponding value of φ reflects H .The detailed discussion of the dynamics for small χ re-quires an understanding of the kinetic term. Its qualitativebehavior depends on σ . For σ < ∼ ( K + 6) χ as compared to 2 Cφ . In theregion of small enough χ the kinetic term becomes thenblock diagonal L ′ kin = 3 M φ ∂ µ φ∂ µ φ + M Cφ ∂ µ χ∂ µ χ. (D.19)In this form we can find a field basis with canonical kineticterms rather easily.For φ > ϕ = r M ln( φ/µ ) , ˜ χ = M √ Cφ χ, (D.20)such that the kinetic term (D.19) takes a canonical form, L ′ kin = 12 ∂ µ ϕ∂ µ ϕ + 12 ∂ µ ˜ χ∂ µ ˜ χ, (D.21)with mixed derivatives ∼ ˜ χ∂ µ ϕ∂ µ ˜ χ again subleading. Interms of these fields the potential reads V ′ = M C + M C " (cid:16) − r ϕM (cid:17) − ˜ χ . (D.22)For ϕ/M < p / ∼ ˜ χ is pos-itive, while it turns negative for larger values of ϕ . Thecosmologies discussed in the main text, where the solution H = H , χ = 0 is unstable such that any perturbation in χ increases, correspond to a negative quadratic term.For σ > ∼ ( K + 6) χ in eq. (D.8) divergesfor χ →
0. The kinetic term takes now for χ → L ′ kin = 3 M φ ∂ µ φ∂ µ φ + M m σ C φ χ − σ ∂ µ χ∂ µ χ. (D.23)The field basis with canonical kinetic terms retains eq.(D.20) for ϕ , while the other degree of freedom is givenby ˆ χ = M m σ | Cφ | (4 − σ ) χ − σ . (D.24) For σ > χ correspond to large negativevalues of ˆ χ , similar to χ R in eq. (127), while for 2 < σ < χ → χ →
0. The potential (D.17) reads V ′ = M C + M C (cid:12)(cid:12) C (4 − σ ) (cid:12)(cid:12) − σ (cid:16) µm (cid:17) σ − σ (D.25)exp σ − σ r ϕM ! " − r ϕM ! − ˆ χM (cid:12)(cid:12)(cid:12)(cid:12) − σ . The stability in the ˆ χ -direction depends on ϕ in a waysimilar to the case σ <
2, but the dependence on ˆ χ is nolonger harmonic. For the example σ = 3 the ˆ χ -dependentpart is quartic, ∆ V ′ ∼ ˆ χ . For σ > χ .
2. Periodic crossing of “big bang singularity”?
In the Einstein frame an oscillation of χ corresponds toa periodic crossing of the big bang singularity. One maywonder if such a behavior is possible within our model. Forthis purpose it is instructive to discuss solutions where ϕ issmall enough such that χ = 0 becomes attractive as timeincreases. We focus on σ < χ . We are interested to find out under which con-ditions there are oscillations around ˜ χ = 0. For this pur-pose we investigate cosmological solutions that are speci-fied by fixing ˜ χ, ˙˜ χ, ϕ and ˙ ϕ at some time t . We choose so-lutions for which ˜ χ ( t ) /M ≪
1, such that the approxima-tion (D.17) is valid, for example ˜ χ ( t ) = 0 , ˙˜ χ ( t ) = c M .We also consider 0 < φ ( t ) < µ , corresponding to nega-tive ϕ ( t ), and take ˙ ϕ ( t ) = 0. For not too large c thepotential V ′ is essentially constant, corresponding to anexponential expansion of the scale factor with H = ± M √ C . (D.26)The scalar field ˜ χ could oscillate around zero, according tothe effective field equation¨˜ χ + 3 H ˙˜ χ + ˜ m ( ϕ ) ˜ χ = 0 , ˜ m ( ϕ ) = M C " exp (cid:16) − r ϕM (cid:17) − , (D.27)provided that ˜ m changes only slowly on the time scale ofthe inverse frequency. One obtains˜ χ = c M ω exp (cid:18) − H ( t − t )2 (cid:19) sin (cid:0) ω ( ϕ )( t − t ) (cid:1) ,ω ( ϕ ) = M C " exp (cid:16) − r ϕM (cid:17) − . (D.28)with ω = ω ( ϕ ) = ω (cid:0) ϕ ( t ) (cid:1) . For negative ϕ these oscilla-tions are fast compared to the Hubble parameter ω H = 24 exp (cid:16) − r ϕM (cid:17) − . (D.29)They are damped for positive H and increase for negative H .5During the oscillations the average value of ˜ χ is givenby h ˜ χ i = c M ω exp (cid:0) − H ( t − t ) (cid:1) . (D.30)Inserting this into the potential (D.22) yields an effectiveexponential potential for ϕ which can be approximated forsufficiently negative ϕ by V ′ eff ( ϕ ) = M C + c M " − r ϕ − ϕ M − H ( t − t ) . (D.31)The field equation for ϕ ,¨ ϕ + 3 H ˙ ϕ = c M √
24 exp − r ϕ − ϕ M − H ( t − t ) ! , (D.32)can be solved approximately by neglecting the term ∼ ¨ ϕ .This results in a relative change of ˜ m ( ϕ ), ∂ t ln ˜ m ( ϕ ) = − √ Cc ( t ) M − r ϕ − ϕ M ! − H,c ( t ) = c exp (cid:0) − H ( t − t ) (cid:1) . (D.33)For consistency of our computation of oscillations the rel-ative change per oscillation period, ω − ∂ t ln ˜ m , has to besmall compared to one. Near t it is indeed small providedthat c is small enough. We may take ϕ ≈ ϕ and useexp( ϕ / √ M ) ≪ √
3, as required by ω ≫ H . For a givenvalue of ϕ one has an upper bound on c such that oscil-lations take place. This bound gets higher if ϕ moves tosmaller values.Our discussion shows that a periodic crossing of the bigbang singularity may indeed occur in our model. It is notclear under what circumstances an initial oscillatory behav-ior can turn into a realistic cosmology at later times. Forpositive H the amplitude of the oscillation of ˜ χ decreaseswith increasing time. If this behavior continues scale sym-metry is asymptotically realized for large t and all particlesare massless. The oscillatory behavior with decreasing am-plitude could be stopped, however, if ϕ increases towardspositive values such that V ( ϕ, ˜ χ ) becomes unstable in the˜ χ -direction. Appendix E: Variable gravity with addi-tional constant coefficient of curvature term
In this appendix we investigate a general class of variablegravity models with effective actionΓ = Z x p ˜ g n − ξ ( ˜ χ + ˜ m ) ˜ R + ˜ K ( ˜ χ )2 ∂ µ ˜ χ∂ µ ˜ χ + ˜ V ( ˜ χ ) o , (E.1)with constant coefficients ξ and ˜ m . This addresses thequestion to what extent the omission of a curvature term with constant coefficient in the action (2) is a crucial in-gredient for our scenario or rather a matter of convenience.For easy comparison we will map the action (E.1) to thekinetial crossover form (2) by suitable field transforma-tions. In particular, we will find that the simple caseof a constant potential ˜ V ( ˜ χ ) = ˜ V or, more generally,˜ V ( ˜ χ → → ˜ V , can realize a crossover scenario very sim-ilar to the one discussed in the main text.An important feature of the action (E.1) is most eas-ily understood in the Einstein frame, where the potentialbecomes V E ( ˜ χ ) = M ˜ V ( ˜ χ ) ξ ( ˜ χ + ˜ m ) . (E.2)Depending on the form of ˜ V ( ˜ χ ) the potential V E ( ˜ χ ) maynot be a monotonic function. For the example ˜ V = µ ˜ χ one finds a maximum of V E for ˜ χ = ˜ m . Such models canstill provide for a satisfactory description of inflation if theinitial value of ˜ χ is close to the maximum of V E . Nev-ertheless, such a setting destroys somewhat the simplicityand beauty of a crossover from a fixed point at ˜ χ = 0 toanother one for ˜ χ → ∞ .For a large class of potentials ˜ V no maximum of V E occurs and V E ( ˜ χ ) is a monotonic function. For example,this happens for a constant ˜ V ( ˜ χ ) = ˜ V or for ˜ V = ˜ V + µ χ if ˜ V is large enough, cf. ref. [3]. We will see that for thisclass of models the kinetial crossover is very similar to theone discussed in the main text.Using the transformation˜ g µν = χ ξ ( ˜ χ + ˜ m ) g µν = χµ p ˜ V g µν , ˜ χ = p ˜ V χξµ − ˜ m , (E.3)one finds indeed the effective action (2), where B = 32 + ˜ K ˜ Xξ + 38 ˜ A ! ˜ D + 32 ˜ A ˜ D, (E.4)with ˜ A = ∂ ln ˜ V∂ ln ˜ χ , ˜ X = ˜ χ ˜ χ + ˜ m = 1 − ξµ ˜ m p ˜ V χ , ˜ D = ∂ ln ˜ χ∂ ln χ = 24 ˜ X − ˜ A . (E.5)To be specific, we take ˜ V = ˜ µ − ˜ A ˜ χ ˜ A (E.6)with constant ˜ A .For 0 < ˜ A < D and therefore B be-comes singular for ˜ X c = ˜ A/
4. This singularity correspondsto the maximum of the potential in the Einstein frame V E ∼ ˜ V / ( ˜ χ + ˜ m ) . One may employ the freeze frame forsolutions where ˜ X = ˜ X c and note the natural occurrence6of large values for B for ˜ X near ˜ X c . Inflationary models ofthis type have been discussed in ref. [68].We observe that B vanishes for ˜ χ → , ˜ X →
0, accordingto B = 32 A ˜ D ! + ˜ K ˜ X ˜ D ξ =
24 ˜ X + 4 ˜ K ˜ Xξ ! (4 ˜ X − ˜ A ) − . (E.7)The limit ˜ χ → χ → ∞ , χ = ξµ ˜ A − ˜ χ − ˜ A ( ˜ χ + ˜ m ) . (E.8)Using ∂ ln ˜ X∂ ln χ = 4(1 − ˜ X )4 ˜ X − ˜ A , (E.9) ∂ ln B∂ ln ˜ X = 1 + 6 ˜ X X + ˜ K/ξ − X X − ˜ A , (E.10)we may compute σ = − ∂ ln B∂ ln χ . For ˜ χ → K ( ˜ χ → > σ = 4 / ˜ A , which increases to σ = 8 / ˜ A if ˜ K ( ˜ χ →
0) = 0.Solutions where ˜ χ approaches zero for increasing time donot provide for an acceptable description of dark energy.On the other hand, for ˜ A < χ → ∞ corresponds to χ → ∞ . In this limit one has ˜ X → , ˜ D → − ˜ A and therefore B = 4(6 + ˜ K/ξ )(4 − ˜ A ) . (E.11)Small values of B , as required for realistic dark energy, canbe achieved if ˜ K ( ˜ χ → ∞ ) reaches − ξ or a value slightlylarger. Thus realistic cosmologies correspond to solutionswhere ˜ χ increases from a value close to ˜ χ c = m q ˜ A/ (4 − ˜ A )to infinity.For a constant ˜ V , i.e. ˜ A = 0, one finds B = 32 + ˜ K ξ ˜ X . (E.12)The freeze frame is valid for the range χ > χ c , χ c = ξµ ˜ m p ˜ V , which corresponds to ˜ χ >
0. Again, one findslarge B in the vicinity of the divergence for ˜ χ →
0, andcorresponding models of inflation [4]. The leading behaviorof B for ˜ χ → B = ˜ K ξ ˜ m + ˜ χ ˜ χ = ˜ K ξ χχ − χ c ,χ c = ξµ ˜ m ˜ V . (E.13) The corresponding value for σ diverges for χ → χ c , σ = − ∂ ln B∂ ln χ = χ c χ − χ c − ∂ ln ˜ K∂ ln χ = χ c − χ ∂ ln ˜ K∂ ln ˜ χ χ − χ c . (E.14)For χ > χ c it drops rather rapidly, however. Estimating forconstant ˜ K the value σ ( N ) at horizon crossing accordingto eq. (48) yields σ ( N ) = (cid:18) ζN ˜ K − (cid:19) − . (E.15)For suitable values of ˜ K/ξ one may obtain values of σ ( N )comparable to the ones discussed in the main text even forconstant ˜ K . On the other hand, for ∂ ln ˜ K/∂ ln ˜ χ = − ˜ σ and horizon crossing in the region χ ( N ) ≫ χ c , one finds σ = ˜ σ . (E.16)This value constitutes a lower bound for σ .The properties of the UV-fixed point for ˜ m > V differ from the ones discussed in sect. II. For ξ ˜ m = c µ and ˜ V = c µ the dimensionless ratios c and c take finite constant values for µ/χ → ∞ . This type ofUV-fixed point resembles the one found in the flow of theeffective average action [12, 13] if one identifies µ with therenormalization scale k .For ˜ A < V diverges for ˜ χ →
0. The limit˜ χ → χ →
0, while for ˜ χ → ∞ one has χ → ∞ . This setting is very close to the one discussed inthe main text. A divergence of B for χ → K for ˜ χ → χ − . Large valuesof B arise rather naturally for small χ if | ˜ A | is small. Onthe other hand, small values of B for χ → ∞ occur if˜ K ( ˜ χ → ∞ ) /ξ is close to − χ requiresa kinetial close to the stability bound at ˜ K/ξ = −
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0. A variation of the kinetial ˜ K ( ˜ χ )is therefore always required for realistic models. The largevalues of B ( ˜ χ →
0) (or B ( ˜ χ → ˜ χ max ) in case of a maximumof V E ) needed for inflation can partially be induced as aneffect of the term ∼ ˜ m R . The crossover scenario discussedin the main text is realized if ˜ V ( ˜ χ →
0) diverges, ˜
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