Inflation, quintessence, and the origin of mass

In a unified picture both inflation and present dynamical dark energy arise from the same scalar field. 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 the exact scale symmetry generates the particle masses. The cosmological solution can be extrapolated to the infinite past in physical time - the universe has no beginning. This is seen most easily in a frame where particle masses and the Planck mass are field-dependent and increase with time. In this"freeze frame"the Universe shrinks and heats up during radiation and matter domination. In the 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 $\Lambda$CDM predicts for the primordial fluctuations a relation between the tensor amplitude $r$ and the spectral index $n,r=8.19(1-n)-0.137$. The crossover is completed by a second stage where the beyond-standard-model sector undergoes the transition to the future fixed point. The resulting increase of neutrino masses stops a cosmological scaling solution, relating the present dark energy density to the present neutrino mass. At present our simple model seems compatible with all observational tests. We discuss how the fixed points can be rooted within quantum gravity in a crossover between ultraviolet and infrared fixed points. Then quantum properties of gravity could be tested both by very early and late cosmology.


I. Introduction
A scalar field plays a dominant role both for inflation in primordial cosmology and dynamical dark energy in the present epoch. The potential of this field constitutes primordial or late dark energy, driving an accelerated expansion in the big bang picture. Quintessential inflation [1,2] identifies the inflaton for inflation and the scalar field of quintessence or cosmon for present dynamical dark energy. In particular, cosmon inflation [3] formulates this unification in the context of variable gravity [4], where the strength of gravity depends on the value of the cosmon field.
Both inflation and quintessence can be closely related to approximate dilatation or scale symmetry. For inflation this symmetry is at the origin of the observed approximate scale invariance of the spectrum of primordial fluctuations. For present dynamical dark energy the cosmon plays the role of the pseudo Goldstone boson of spontaneously broken dilatation symmetry [5]. In case of exact dilatation symmetry it would be an exactly massless dilaton, while a tiny mass and potential are generated by a "scale symmetry violation" or "dilatation anomaly". Scale symmetry is intimately related to fixed points of "running" dimensionless couplings or mass ratios. At a fixed point any information about intrinsic mass or length scales is lost. Quantum scale symmetry is then realized even if the underlying quantum field theory is not scale invariant.
The presence of approximate scale symmetry both for the primordial and late cosmology suggests that the infinite past and infinite future of the universe correspond to fixed points. We propose here that the two fixed points have different properties. For the fixed point in the infinite past scale symmetry is not spontaneously broken. All masses vanish. In contrast, the fixed point that will be approached in the infinite future is characterized by spontaneous symmetry breaking of dilatation symmetry, resulting in a spectrum of massive particles and a massless dilaton. The way how scale symmetry is realized and explicitly or spontaneously broken is directly related to the origin of mass. All particle masses are generated either by explicit or spontaneous breaking of scale symmetry. The explicit breaking by intrinsic mass scales plays a crucial role for the crossover between the two fixed points. Spontaneous breaking characterizes the "future fixed point" and our present universe. The basic mechanisms that generate the particle masses provide the physical "raison d'être" for inflation and late dark energy, such that these key cosmological ingredients appear less "ad hoc".
Inflation describes the vicinity of the past fixed point. The inflationary epoch has to end, however. "Late cosmology" comprises epochs of radiation-, matter-and dark energy domination. It is characterized by the approach towards the future fixed point. We will describe the transition from inflation to late cosmology as a first stage of the crossover between the two fixed points. In the crossover region couplings have to run from one fixed point to the other. Scale symmetry is therefore necessarily violated in the crossover region. This is the basic reason for the qualitative change in the dynamics of the cosmon that occurs at the end of inflation.
If there is more than one relevant or marginal deviation from the "past fixed point" the crossover may occur in different stages. In case of a slow running (e.g. logarithmic) the scales associated to these stages can be separated by many orders of magnitude. We assume here that in the singlet sector of the standard model of particle physics the crossover is completed only in a second stage. This singlet sector influences the masses of the neutrinos by "nonrenormalizable operators" according to the see-saw or cascade mechanism. While the mass ratios of all particles except for neutrinos reach fixed values already at the end of inflation, the ratio of neutrino mass to electron mass makes the transition to the future constant value only in the present epoch. The relative increase of the neutrino masses realizes "growing neutrino quintessence" [6,7] and explains the "why now problem" by relating the present dark energy density to the present neutrino mass.
The history of dark energy reflects the two stages of the crossover. A primordial scaling solution corresponds to dominant dark energy during inflation. The first stage of crossover ends this scaling solution, triggering a transition to a different scaling solution during the radiation and matter dominated epochs. As a consequence of this scaling solution dark energy decreases proportional to the dominant radiation or matter component [5], constituting a small fraction of "early dark energy". Neutrinos are relativistic during this epoch and their masses play no role. The second stage of the crossover takes place in the present cosmological epoch. A substantial increase of the neutrino masses ends the second scaling solution once neutrinos become non-relativistic. This cosmic "trigger event" has happened around redshift z ≈ 5, inducing a transition epoch with dominant dark energy and accelerated expansion. Once the second stage of the crossover is completed, the ratio between neutrino and electron mass approaches a constant value according to the future fixed point. Cosmology in the far future may correspond to a new scaling solution for which dark energy needs not to remain dominant.
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 "standard model fixed point". For this fixed point the renormalizable dimensionless couplings of all particles are the ones observed in present experiments. Neutrino masses, however, are typically substantially smaller than their present value. The standard model fixed point is unstable in the sector of heavy singlets, which will finally drive the flow trajectory away from it and towards the infrared fixed point. On the other hand, the zeros of the β-functions for the renormalizable couplings of the standard model are stable for decreasing µ, such that the presently measured values hold to high accuracy for the entire matter and radiation dominated epochs. The second step of the crossover affects first only the neutrino masses. Nevertheless, when the second step of the crossover will be completed in the far future, it is possible that the changes in the singlet sector also affect the renormalizable couplings of the standard model. Their values at the infrared fixed 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 shows the first stage of the crossover from the UV-fixed point to the vicinity of the standard model fixed point, and the subsequent second step of the crossover to the infrared fixed point. We also associate the different cosmological epochs to the corresponding parts of the flow trajectory.
This paper is organized as follows: In sect. II we introduce the flow equations underlying our approach. They describe the change of couplings as an intrinsic overall mass scale µ is varied, not to be confounded with the momentum dependence of couplings. We discuss the properties of the ultraviolet and infrared fixed points. For this purpose we choose a frame of variable gravity where the crossover . Arrows indicate the direction of decreasing µ or increasing χ. This direction corresponds to the flow of cosmic time from the infinite past (UV) to the infinite future (IR). The crossover trajectory passes near an (approximate) fixed point (SM) that characterizes the present standard model. The two regimes of fast changes, CR1 and CR2, correspond to the two steps of the crossover. We also indicate the corresponding cosmological epochs: inflation (I), end of inflation (EI), radiation domination (R), matter domination (M), dark energy domination (Q).
is described by the flow equation for the "kinetial", e.g. the coefficient of the scalar kinetic term. In particular, we investigate settings where the kinetial diverges at the ultraviolet fixed point with a large anomalous dimension.
In sect. III we turn to the cosmological solution that follows for values of the cosmon field χ close to the ultraviolet fixed point and the first step of crossover away from it. It describes an epoch of inflation and its end. We compute the properties of the primordial density fluctuations. Both the spectral index n and the tensor to scalar ratio r are determined by the anomalous dimension σ and therefore related, 1 − n = r(2 + σ)/16. Relating r to the number of e-foldings we find n 0.967, r 0.13, with values close to the bounds. The crossover provides for a natural explanation of the small amplitude of primordial fluctuations. This amplitude is suppressed by the ratio of the intrinsic mass scale µ over the crossover scale m, which is exponentially small due to the slow running near the fixed point.
Sect. IV discusses "late cosmology" after the end of inflation. It starts with a scaling solution for the radiation and matter dominated epochs that is characterized by a small almost constant fraction of early dark energy [5,[8][9][10]. This scaling explains why the present dark energy density is of the same order as the present matter energy density. In particular, we discuss models where the infrared fixed point corresponds to a "conformal kinetic term". The deviation from the fixed point is characterized by a function B(χ/µ) that decreases with an inverse logarithm for large χ/µ, B −1 = κ ln(χ/µ). The fraction in early dark energy is proportional to B and therefore naturally small for the large values of χ relevant for late cosmology. The slow flow of B induces small scaling violations for the cosmological solution that we discuss in terms of an approximate analytic solution. We find a lower bound of the dark energy fraction at last scattering, Ω ls h 0.012, close to the observational bounds. As the second step of the crossover sets in the neutrino masses start to increase substantially. Once neutrinos become non-relativistic they stop the scaling solution, "freezing" the dark energy density at the value it has reached at this moment. This leads to a phenomenology very close to a cosmological constant, with a value determined by the present average neutrino mass. Such a scenario solves the "why now?" problem.
In sect. V we turn more closely to the particle physics aspects of the ultraviolet fixed point. For an anomalous dimension in the range 1 < σ < 2 the couplings of the renormalized cosmon field are asymptotically free. The assumed fixed point that would provide for non-perturbative renormalizability (asymptotic safety) of quantum gravity has then the simple structure of a massless renormalized scalar field (with standard kinetic term) coupled to fourth order gravity. The non-perturbative character is related to anomalous dimensions for deviations from this fixed point that are of the order one. We argue that large anomalous dimensions can lead to a natural explanation of the small ratio Fermi scale/Planck scale and therefore provide for a possible solution of the gauge hierarchy problem. The gauge hierarchy and the small amplitude of primordial fluctuations become related.
Sect. VI describes the ultraviolet fixed point in different frames (different choices of field variables for metric and scalar field). We show that our ansatz with a simple quadratic cosmon potential and crossover described by the kinetial can be obtained by field transformations from a very large class of variable gravity models. It is therefore rather generic. The description of the crossover in terms of the kinetial is a convenience rather than a fundamental future. Our conclusions are presented in sect. VII.

II. Fixed points and crossover
In quantum field theories the renormalized dimensionless couplings "run" as functions of an intrinsic mass scale µ. Here we consider all intrinsic mass parameters as being proportional to µ, with ratios of intrinsic mass scales associated to dimensionless couplings. For a fixed point this flow stops and dimensionless couplings become independent of µ. An ultraviolet (UV) fixed point is reached if suitable dimensionless couplings reach constant values for µ → ∞. Such a fixed point renders gravity nonperturbatively renormalizable (asymptotic safety [11][12][13][14]). Dilatation symmetry is an exact quantum symmetry at the UV fixed point. An infrared (IR) fixed point corresponds to the stop of the flow of dimensionless couplings for µ → 0. All intrinsic mass parameters vanish in this limit. With dimensionless couplings independent of µ scale symmetry is again realized. In general, the existence of an IR fixed point is not compulsory -alternatives are diverging dimensionless couplings for µ → 0 or even a breakdown of the model at a critical value µ c > 0. We assume here that such divergencies do not happen and an IR-fixed point therefore exists. A first functional renormalization investigation of such a possible IR fixed point can be found in ref. [14].
The flow of couplings as a function of µ should not be confounded with the running as a function of momentum divided by particle mass (say the electron mass). It is this running as a function of momentum/mass that is described by the usual β-functions of the standard model of particle physics. In contrast, the flow as a function of µ describes the effect of a simultaneous change of all intrinsic mass scales ∼ µ. The µ-flow equations need a separate computation which has not yet been performed. They are similar in spirit to the running of couplings as a function of a mass parameter investigated by Symanzik [15]. Different µ correspond to a family of different theories. There is also some analogy to the functional renormalization flow of the effective average action [16,17], with IR-cutoff k associated to µ. We emphasize, however, that µ does not act as an effective cutoff for all modes -the graviton is massless for all µ.

Variable gravity
We will work within variable gravity [4] and investigate the cosmological solutions of the field equations derived from the quantum effective action for the coupled cosmongravity system (1) The variable Planck mass is given by the value of the cosmon field χ. The quadratic cosmon potential V = µ 2 χ 2 involves the intrinsic mass scale µ. A large family of effective actions can be brought by field transformations to a form where the coefficient of the curvature scalar R is − 1 2 χ 2 and the scalar potential is quadratic, V (χ) = µ 2 χ 2 . We will discuss this issue in sect. VI. We then remain with the dimensionless function B(χ/µ). Its dependence on µ is described by the µ-flow equation. Stability requires B > 0.
The quantum effective action should be supplemented by higher order curvature invariants, These terms will play a role for graviton-graviton scattering at the UV-fixed point and for the approach of the cosmological solution to the infinite past, χ/µ → 0. For the cosmological epochs discussed in this paper they are subleading and will be omitted in the explicit calculations. We do not include a possible scale invariant contribution to the cosmon potential ∆V = λχ 4 . Indeed, the functional renormalization investigation [14] of the behavior of a possible fixed point suggests that the cosmon potential cannot increase ∼ χ 4 for χ → ∞.
Dimensionless functions as B (or C and D) can only depend on the dimensionless ratio χ/µ. This links their µ-dependence according to the flow equation to their dependence on χ. The UV-fixed point for µ/χ → ∞ can also be seen as the limit χ → 0, while the IR-fixed point corresponds to the limit χ → ∞. We will find cosmological solutions where χ varies from χ → 0 in the infinite past to χ → ∞ in the infinite future. This is how cosmology can describe the crossover between two fixed points. The UVfixed point for χ → 0 will often be called the "past fixed point", and the IR-fixed point for χ → ∞ is associated with the "future fixed point".

Infrared and ultraviolet fixed points
For the IR-fixed point µ vanishes and B reaches a constant lim µ→0 B(χ/µ) = B ∞ . The term ∼ µ 2 χ 2 in eq. (1) is absent for µ → 0, and the quantum effective action contains no longer any parameter with dimension mass. It is invariant under scale symmetry, with a scaling of χ according to its canonical dimension.
For the realization of an UV-fixed point the anomalous dimension of the cosmon will be crucial. Indeed, for a canonical scaling of χ the "mass term" ∼ µ 2 χ 2 would spoil scale invariance for µ → ∞. An anomalous dimension for µ → ∞ is realized if B(χ/µ) diverges for µ → ∞ with a power law, For σ < 2 the gravitational higher order invariants (2) and the scalar kinetic term are then invariant under the scaling At the UV-fixed point the effective action can be written in terms of a renormalized field, as For constant C and D scale invariance is manifest -it is the renormalized field that shows the standard scaling χ R → α −1 χ R . The remaining terms for the cosmon potential and ∼ R, vanish for µ → ∞ and fixed χ R , provided σ > 1. In this case ∆Γ UV accounts for deviations from the fixed point and can be neglected at the U V -fixed point. At the fixed point one finds a free massless scalar field coupled to higher order gravity. For the boundary case σ = 1 the renormalized scalar field has a non-vanishing self-interaction, with scale invariant potential Also the limiting case σ = 2 can be associated with a fixed point -see sect. V for a more detailed discussion. An UV-fixed point is therefore realized for Besides the cosmon-gravity part of the effective action (1), (2) we also have to specify the part for matter and radiation. We will take the standard model of particle physics and assume that all dimensionless couplings (e.g. gauge couplings, Yukawa couplings, Higgs-boson self interaction), normalized at momenta ∼ χ, are functions of χ/µ that reach fixed constant values for µ → ∞ and µ → 0. The UV-values are typically different from the IR-valuesfor example, one may imagine that all renormalized couplings vanish for µ → ∞, leaving only free particles at the UV-fixed point. We write the coefficient of the quadratic term in the Higgs potential as −ǫ H (χ/µ)χ 2 . For an IRfixed point the dimensionless coupling ǫ H goes to a (very small) constant. No memory of the scale µ is then left for χ → ∞ -the Fermi scale is proportional to χ such that the charged lepton and quark masses as well as the gauge boson masses are proportional to χ [5]. For a constant strong gauge coupling g s (p 2 ∼ χ 2 ) also Λ QCD scales ∼ χ, such that hadron masses are ∼ χ as well. On the opposite end we assume that the renormalized coupling corresponding to ǫ H does not diverge for χ → 0, such that all particles are massless at the UV-fixed point.
The same general picture applies for particles beyond the standard model, in particular the sector of heavy singlets which influence the neutrino masses by the seesaw [18][19][20] or cascade [21][22][23][24] mechanism. The only difference to the standard model sector will be the relevant value of χ/µ for which the crossover between the two fixed points takes place.
The cosmology at the fixed points is not per se very interesting. For the past fixed point matter and radiation may be negligible. The field equations for the cosmongravity system derived from the effective action (1), (2) admit the simple flat space solution The cosmology for the future fixed point is again of a simple type. A scale invariant model that obtains by omitting in eq. (1) the potential µ 2 χ 2 has been proposed by Fujii [25,26]. After Weyl scaling it describes standard cosmology plus a massless dilaton with derivative couplings. The dilaton settles to a fixed value after a short period of initial damping of its motion, and plays no role for the subsequent "late" cosmology of the present epoch [5]. While interesting in its own right, such a scale invariant model cannot describe dynamical dark energy or quintessence. In our setting the cosmology of this model is reached for the future fixed point.
The interesting cosmological features of inflation and dynamical dark energy are a consequence of small scaling violations in the vicinity of the fixed points. Close to the past fixed point the scale symmetry violating terms ("dilatation anomaly") ∆Γ UV , cf. eq. (7), render the cosmological solution (10) unstable such that any small value of χ slowly increases with increasing time t. This slow increase will be associated with the almost scale invariant epoch of inflation. As χ grows large enough the crossover to the future fixed point starts and inflation ends.
The subsequent radiation and matter dominated epochs belong already to the neighborhood of the standard model fixed point. For this fixed point the dominant scaling vi-olation arises from the cosmon potential ∼ µ 2 χ 2 . This will describe dynamical dark energy, according to an approximate scaling solution with dark energy proportional to the dominant radiation or matter component. Indeed, the ratio of the potential V = µ 2 χ 2 divided by the fourth power of the effective Planck mass χ 4 decreases ∼ µ 2 /χ 2 and reaches tiny values as χ moves to very large values in late cosmology. At the second step of the crossover an additional violation of dilatation symmetry in the neutrino sector stops the scaling evolution of the cosmon.
Realistic scaling solutions with a small fraction of early dark energy have been extensively discussed [4,5,8] for the case where B reaches for µ → 0 a small value Observational bounds on early dark energy restrict the allowed values to α 10 [27][28][29][30][31]. While this setting is perfectly viable, we investigate in the present paper the possible alternative that B vanishes at the IR-fixed point, B ∞ = 0. For the "conformal value" B = 0 the cosmon is no propagating degree of freedom. Furthermore, for B < 0 the model becomes unstable. The flow of couplings typically avoids to cross from a stable to an unstable situation. It seems therefore reasonable to assume that B = 0 is a fixed point of the flow of B, and we will assume that it is reached for χ → ∞. For finite χ one has B > 0.

Crossover
The crossover that leads to the end of inflation is related to the flow of the dimensionless function B(χ/µ). We first discuss a simple one-parameter flow equation, corresponding to an anomalous dimension σ = 1, The approach to the fixed point at B = 0 is quadratic (vanishing anomalous dimension) while the approach to the fixed point B −1 = 0 involves an anomalous dimension The finer details of the crossover will be less important. The UV-fixed point B −1 = 0 is approached for µ → ∞ or χ → 0. This fixed point is relevant for the infinite past of our universe. The IR-fixed point B = 0 is approached for µ → 0 or χ → ∞. It governs the infinite future. The solution of eq. (12) involves an integration constant c t which determines the particular trajectory of the flow according to the implicit solution It is related to a mass scale m by dimensional transmutation, m = µ exp(c t ). The crossover between the two fixed points occurs in the region χ ≈ m and we will see that this coincides with the end of inflation. Late cosmology corresponds to χ ≫ m, while primordial cosmology is characterized by χ ≪ m. With B(χ) determined by eq. (15) our model (1) contains two free dimensionless parameters in the scalar-gravity sector, namely κ and c t . We will find below that realistic cosmology is obtained for No tiny or huge dimensionless parameters appear in our setting.
The flow equation (2) is is only a particular example for a crossover between two fixed points for which B −1 or B vanish, respectively. An extended family of models may be given by with solution Eqs. (12), (15) correspond to σ = 1, while eq. (17) accounts for an arbitrary anomalous dimension σ at the UVfixed point, The crossover behavior needs also to be specified for the particle physics sector of our model. We present details in the appendix A and outline here only a few characteristics. We will assume that for the large values of χ/m relevant for nucleosynthesis and later epochs the dimensionless couplings are already very close to their fixed point values, such that their dependence on χ can be neglected for the purpose of cosmology. Similarly, we assume for these periods that the masses of all particles except for neutrinos have reached the scaling behavior m p ∼ χ appropriate for the fixed point. With this simple assumption the severe observational bounds on the time variation of fundamental couplings and apparent violations of the equivalence principle are obeyed [5].
Neutrino masses also involve a heavy singlet sector by virtue of the seesaw or cascade mechanism. For this sector we postulate that the crossover is happening in the region of χ relevant for present cosmology, such that the present variation of the average neutrino mass with χ, involves a parameterγ > 0. This parameter only matters for a rather recent cosmological epoch when neutrinos have become non-relativistic. It plays no role as long as neutrinos are relativistic. Together with the present average neutrino mass m ν0 the five parameters κ, σ, c t , m ν0 and γ, supplemented by the values of particle masses including some dark matter candidate, will be sufficient to describe a realistic cosmological sequence of inflation, radiation-and matter-domination, as well as the present transition to a new dark energy dominated epoch. So far, our model is compatible with all present observations. We will see that the anomalous dimension σ is closely related to the amplitude of primordial gravitational waves. For σ ≤ 2, as appropriate for the existence of an UV-fixed point, we find a tensor amplitude r ≥ 0.13.

Primordial cosmology
We begin with a brief discussion of primordial cosmology in the freeze frame. We omit the higher order curvature invariants (2). As we will see below they affect only the remote past of the universe before observable density fluctuations left the inflationary horizon. We start with σ = 1 and approximate χ ≪ m, B = m/χ. The primordial epoch will correspond to an inflationary epoch in the Einstein frame. For this epoch we neglect matter and radiation. The cosmon field equation obtains by variation of the effective action (1) and reads [4] Here we have inserted the expression for R according to the gravitational field equation. The Hubble parameter is given by We may use dimensionless variables y = mt, w = χ/m, h = H/m, λ = µ 2 /m 2 , such that The approximate solution for w → 0 in next to leading order approaches a constant h with w 0 vanishing in the infinite past for y → −∞, and y c ,w 1 integration constants. Restoring dimensions yields in leading order .
We conclude that time can be continued in this approximation to the infinite past, t → −∞. In this limit geometry approaches de Sitter space and the cosmon field vanishes. The limiting solution H = µ/ √ 3, χ = 0 is unstable, however. A small deviation χ increases with t according to eq. (26) or (24), and the Hubble parameter decreases. Primordial cosmology describes an inflationary epoch. This will end if the increase of χ or decrease of H becomes too fast. A quantitative estimate for the end of inflation will be given later in the Einstein frame.
For σ = 2 we use B = m 2 /χ 2 , such that the field equations take the form The leading order solution becomes now For 1 < σ < 2 one finds a decrease of χ towards the infinite past with an intermediate power of inverse time.
We will see below that the solutions (26), (28) correspond to a standard inflationary scenario in the Einstein frame or "big bang frame". Horizon crossing of the observable primordial fluctuations occurs when χ/µ is already large, χ ≈ 1.5 · 10 4 µ. For these values the relative contribution of higher order curvature invariants from eq. (2) is suppressed by a factor ∼ Cµ 2 /χ 2 , which is tiny for any moderate C. We can therefore neglect such terms for the discussion of observable signals from inflation. Nevertheless, as χ becomes much smaller than µ for t → −∞, the role of the higher order curvature invariants becomes important. Typically, the geometry will approach flat space in the infinite past. We will give more details of the behavior of cosmology in the infinite past in a separate work.
We will show next that our crossover model predicts a rather large ratio between primordial tensor and scalar fluctuations, r 0.13. Since the value of the cosmon field χ (which plays the role of the inflaton) equals the dynamical Planck mass, the Lyth bound [32,33] plays no role in our setting of variable gravity [34].

Cosmon inflation
The association of primordial cosmology with an inflationary epoch is most easily understood in the Einstein frame. Also a quantitative discussion of the generation of primordial density fluctuations and the end of inflation is best done in this frame. With the quantum effective action (1) reads We identify M = 2.44 · 10 18 GeV with the Planck mass and observe that the cosmon potential V ′ decays exponentially to zero [5,8]. The kinetial k is related to B by We choose α such that the field ϕ has a standard normalization for the present cosmological epoch, k 2 (ϕ 0 ) = 1, or 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. A slow roll period for inflation is realized for large enough k 2 . We consider here a general function B(χ) and specialize to eq. (19) later. The usual slow roll parameters ǫ and η obtain as [3,4] Inflation ends when ǫ or |η| are of order one. We take B(χ f ) = 6 with ǫ f = 1/3, η f = 2/3−σ/6. This is the value where the kinetic term in eq. (1) changes sign. Inflation is realized for a rather generic shape of the function B(χ). It is sufficient that B is large enough for small χ in order to induce an epoch of slow roll, and that B − 6 reaches negative values as χ increases in order to end inflation. The definition of σ employed in the present section, given by eq. (33), differs slightly from the preceding section. In the present section, σ is considered as a function of χ. It agrees 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 use of the same symbol.

Spectral index and tensor ratio of primordial fluctuations
The spectrum of primordial scalar density fluctuations is characterized by the spectral index n = 1 − 6ǫ + 2η, while the relative amplitude of tensor fluctuations over scalar fluctuations reads r = 16ǫ. Here ǫ and η have to be evaluated for the value of χ at horizon crossing, N e-foldings before the end of inflation. One finds the relations We observe an interesting general relation between n and r. Horizon crossing occurs in the region χ ≪ m , B ≫ 6. The particular models with σ = 1 or σ = 2 predict For a spectral index n = 0.965 this implies a rather high amplitude, r = 0.19 or 0.14, in the range claimed by BICEP [35].
We next compute the relation between the value of χ(N ) at horizon crossing and the number N of e-foldings before the end of inflation, In the range of interest B(χ) is typically a strongly decreasing function. The integral is dominated by the region around χ(N ) where we may With B(χ f ) = 6 one finds and These two central formulae express both n and r in terms of σ(N ) and N . For the particular model with σ = 1 one obtains B(N ) = 2N + 6 and predicts while σ = 2 yields We show the spectral index and the tensor ratio as a function 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 of the entropy production after the end of inflation. A typical value is N = 60. For a given N both n and r are uniquely Thus only the logarithmic derivative of ln B at a particular value of B matters for the computation of n and r!

Amplitude of primordial fluctuations
The amplitude of the primordial scalar fluctuations can be related to the value of the potential at horizon crossing and the tensor to scalar ratio where the last equation employs the observed amplitude of the spectrum of CMB-anisotropies. This measurement determines the ratio .
We next employ the approximate form B = (m/χ) σ or such that For the particular model with σ = 1 one finds For the constant c t in eq. (15) one infers (For the numerical value we have taken N = 60, see below.) Due to the exponential dependence on c t no very large or small parameter is needed in order to obtain a small fluctuation amplitude The flow equation (12) generates the scale m by dimensional transmutation. The small amplitude A indicates that this scale is larger than the "intrinsic scale" µ. The situation is similar for other values of σ. For σ = 2 the ratio m/µ decreases by a factor 1/ √ 30 as compared to σ = 1. We may turn this argument around and state that crossover models provide for a natural explanation of a small fluctuation amplitude A. We can use the dimensionless cosmon potential V /µ 4 = χ 2 /µ 2 in order to define the dimensionless flow parameter bỹ We have associated the scale m with the crossover value µ cr where the flow moves away from the behavior dictated by the "past fixed point" forμ → ∞, Different trajectories (solutions of the flow equations) can be characterized by how close to the fixed point they are forμ = 1. The larger m/µ, the closer a trajectory is to the fixed point. In view of the exponential behavior of eq. (51), already moderate negative values ofμ cr are sufficient to induce large values of m/µ, and therefore a small amplitude A ∼ (µ/m) 2 ∼ e 2μcr . We may also evaluate the dimensionless ratio For χ = m this quantity measures the potential in units of the Planck mass at the crossover. For many of our models the corresponding scale of the potential in the Einstein frame V 1 4 is of the order where spontaneous symmetry breaking is expected in a grand unified theory. This suggests that the crossover could be associated with grand unified symmetry breaking.
Finally, we may compare the value of χ(N ) at horizon crossing with m using eq. (45), For all models (1 ≤ σ ≤ 2) one finds a small value x(N ) ≪ 1, justifying the approximation (3). On the other hand, we observe that χ(N ) is much larger than µ, cf. eq. (44) 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.

Horizon crossing
We next need to evaluate the value of N for our type of crossover models. We present here a detailed treatment that allows one to estimate where various uncertainties come from. Horizon crossing of a mode with comoving wave vector k occurs for where a out or a in corresponds to the scale factor when the mode leaves the horizon after inflation or enters again in the more recent past. We use with a f and a r the scale factors at the end of inflation and at a time when the universe begins to be dominated by radiation, respectively, and H r = H(a r ). For a out /a f = e −N one finds the relation Neglecting entropy production for photons for a > a r we use a in /a r = T r /T in which T the photon temperature. We relate T to the total energy density in radiation 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 We first evaluate N 0 for modes that come into the horizon today, and subsequently extrapolate to larger k. The dominant contribution is the first term ∼ ln(H out /H in ). We can relate H out to the tensor amplitude of the primordial fluctuations and use 3M 2 H 2 in = ρ c = (2 · 10 −3 eV) 4 , which yields With Ω 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. , With eq. (45) one finds or with r depending on N and σ according to eq. (38). Inserting N 0 ≈ 65 in the subleading term ∼ ln r, and κ = 1 2 (see below) yields for σ = 1(σ = 2) The remaining piece ∆N reflects the details of entropy production between the end of inflation (a f ) and the beginning of the radiation dominated universe (a r ). We may parametrize this epoch by two parameters, the number of e-foldings N f r for the duration of this period and the averaged equation of statew which governs the evolution of the total energy density, With one finds For a fast entropy production N f r is of the order one. The parameterw is a suitable average of a function w(a) that starts close to w(a f ) ≈ −1 for a = a f , may then be given for a period of domination of scalar kinetic energy, w(a) ≈ 1, and finally end with w(a r ) ≈ 1/3. For not too large N f r andw close to 1/3 one may simply neglect ∆N , and we will use this approximation in the following. We may finally extrapolate to modes with present wavelength smaller than the horizon. As compared to N 0 the dominant correction factor is with k and L the wave number or wave length of the mode, and index zero denoting the ones corresponding to the present horizon (L 0 ≈ 3000M pc). In the range where primordial gravitational waves may be detected (k/k 0 ≈ 80) one has δN ≈ −4.4, such that a reasonable overall estimate is We can neglect the σ-dependence of N and use N = 60. Up to small calculable corrections for σ = 2 this entails the predictions In particular, one obtains for σ = 2 in accordance with the prediction in ref. [36] (see also refs. [37][38][39]). For σ = 1 one finds r = 0.25, n = 0.953. Due to the dependence of N on the wave number k the spectral index and tensor amplitude depend on k according to Since |∂σ/∂ ln N | is typically of the order one or smaller the running of the spectral index is very slow. Indeed, for our model σ changes over 60 e-foldings only from σ(N ) to and this change occurs towards the end of inflation. We conclude that our UV-fixed point scenario provides for a rather simple and predictive model of inflation.

IV. Late cosmology and dark energy
The crossover in the kinetial K(χ) = B(χ) − 6 from positive to negative values triggers the end of the inflationary slow roll solution. Subsequently, radiation and entropy are produced by various mechanisms [3,37]. We recall that during the crossover the dimensionless couplings and mass ratios of the standard model of particle physics are supposed to change from their values for the past fixed point to the ones for the standard model fixed point. In particular, the effective quartic cosmon-Higgs-coupling ǫ H could be of order one instead of the tiny value for the future fixed point, such that a fluctuating Higgs doublet could play a major role for the heating [3]. More precisely, a χ-dependence of ǫ H results in the Einstein frame in an effective coupling between the cosmon and the Higgs doublet. This generalizes to other χ-dependent dimensionless couplings. In a grand unified theory the heating period may be associated with the onset of spontaneous symmetry breaking of the GUT-gauge group. After the heating and entropy production have occurred the universe enters its "late epoch", beginning with radiation domination. The late universe is characterized by the approach to the future fixed point. During the radiation and matter dominated periods this approach is slow, as accounted for by the (approximate) standard model fixed point, recall Fig. 1.
In the freeze frame the particle masses increase with increasing χ, while the universe shrinks, in contrast to the usual big bang picture [36]. (For early cosmological models with varying particle masses see. refs. [40][41][42].) Indeed, only the dimensionless ratio of the distance between galaxies divided by the atom radius is observable [43][44][45][46]. The overall picture has been described in detail in ref. [4] for the case where the kinetial takes a constant value K ∞ at the IR-fixed point. In the present crossover model B depends only mildly on χ for χ ≫ m, The cosmology with constant B ∞ = B(χ → ∞) = 4/α 2 is therefore a good approximation. A priori, both the behavior (77) and a small fixed value of B ∞ are perfectly viable candidates for realistic late cosmology. In the present paper we supplement the earlier discussion with constant B ∞ by a quantitative investigation of a slowly varying B(χ) according to eq. (77). We employ the Einstein frame in order to facilitate the embedding of this model in standard scenarios of quintessence.

Late cosmology in the Einstein frame
In the Einstein frame (30) the kinetial is given by with ϕ 0 the present value of the cosmon With the exponential potential (30), this is a typical model of dynamical dark energy. Except for neutrinos the standard model particles and dark matter do not couple to ϕ. The cosmon field equation (80) involves the cosmon-neutrino coupling with m ν the ϕ-dependent average neutrino mass. The Hubble parameter obeys with and ρ r,m,ν the energy densities of radiation, matter and neutrinos, respectively. While ρ r and ρ m obey the usual conservation equations,ρ r = −4Hρ r ,ρ m = −3Hρ m , the neutrinos exchange energy momentum with the cosmon due to the variable masṡ (The second equation follows from eqs. (80), (83).) We may follow the evolution in terms of y = ln a + y 0 instead of time [7,47,48], with is the same as in eq. (20). It will determine the precise timing of the crossover to dark energy domination. As long as neutrinos are relativistic one has w ν = 1/3 and the terms ∼γ can be neglected. For the radiation dominated epoch we can neglect ρ m and incorporate ρ ν into ρ r , ρ r =ρ r M 4 exp(−4y). For the matter dominated period neutrinos can be neglected as long as they are relativistic, similar to radiation. We only need to keep ρ m =ρ m M 4 exp(−3y). We will combine the discussion of these periods by taking ρ d =ρM 4 exp(−ny) for the energy density of all other components except the cosmon, with n = 4(3) for radiation (matter) domination.
The last epoch in the cosmological evolution starts when neutrinos become non-relativistic. The terms proportional to the cosmon-neutrino couplings β in eq. (84) can no longer be neglected. For large enoughγ they stop effectively the further change of ϕ, such that V ′ (ϕ) acts like a cosmological constant. This scenario of "growing neutrino quintessence" [6,7,[49][50][51][52] relates the present dark energy density to the average neutrino mass A realistic present dark energy fraction Ω h (t 0 ) ≈ 0.7 is found forγ m ν (t 0 ) = 6.15eV.
This relation remains valid with good accuracy even in presence of the large scale non-linear neutrino lumps that form and dissolve periodically after redshift z ≈ 2 [53].

Approximate analytic solution
For a constant kinetial k = 1 one finds for radiation or matter domination the standard scaling (tracker) solution for quintessence with an exponential potential [5]. For slowly varying k(ϕ) according to eq. (78) we may therefore use the approximation of a solution in the vicinity of the scaling solution. The difference between the cosmon field according to a model with ϕ-dependent kinetial on one side, and the scaling solution on the other side, is denoted by M δ(y).
For the evolution equations for ϕ and ln ρ h we make the ansatz such that For constant k 2 one recovers the scaling solution [5,8,54] with a constant fraction of early dark energy Ω e , δ = 0, ∂ y f = 0,ρ = 6 (6 − n)f , For a smooth enough ϕ-dependence of k 2 we therefore expect a behavior close to this scaling solution. We employ and find with For the variables one obtains At this point we assume that (nM/α)∂ ln k 2 /∂ϕ is small. We can then expand in small ∆ and u, With ∂ ln k 2 /∂ϕ and Ω h varying slowly the solution approaches the particular approximate solution Indeed, if we neglect the y-dependence of∆ andū one has for u ′ = u −ū, ∆ ′ = ∆ −∆ the linear evolution The eigenvalues of the stability matrix A are both negative, implying an exponential decrease of ∆ ′ and u ′ as y increases. We conclude that cosmology approaches a solution with non-vanishing early dark-energy fraction decreasing with decreasing k 2 , Solving eqs. (100),(98) for Ω h we end with a general formula for slowly varying k(ϕ), Here we recall that for generic models of quintessence the formulation with exponential potential (30) and possibly varying kinetial k(ϕ) can be obtained by an appropriate rescaling of the scalar field. Let us now turn to our model with k 2 = M α/ 2κ(ϕ−φ) and One hasū and for Ω h ≪ 1 a small parameter κ ≪ 1 indeed implies u ≪ 1. In turn the early dark energy fraction decreases logarithmically for increasing χ.

Bounds on parameters
Besides the determination ofγ by a measurement of the present dark energy fraction and the average neutrino mass (88) we can use bounds on early dark energy for an estimate of the parameter κ.
If we require κ < 1/2 in order to maintain smallū this yields a dark energy fraction larger than 2% which could be detectable in the future [8,55,56].
For the present epoch one has Interestingly, for κ < 1/2 one finds an upper bound on α, α < 15.6. Over the restricted range since last scattering k 2 has changed only little such that at last scattering the relation Ω h ≈ 3/α 2 1/80 is valid, For κ < 0.5 the model therefore predicts a lower bound on the fraction of dark energy at last scattering This is at the borderline of a possible detection with present observations [27][28][29][30][31]57]. One therefore infers the bound κ 0.5. Formally, we may combine eqs. (100), (103), (108) in order to obtain for k 2 ≈ 1, n ≈ 3, Ω h ≪ 1 the relation This would imply a minimal value for Ω h . The minimum is reached, however, only for large κ for which our approximation no longer holds. We will consider in this paper a value κ = 0.5 which is compatible with observation and consistent with our approximate solution with smallū.
In summary, the late cosmology of our model resembles closely growing neutrino quintessence with a variable cosmon-neutrino coupling β [7]. The interesting new features are an explanation of a large effective value for α in terms of the approach to the IR-fixed point, and the association of large positiveγ = −β/α with a crossover affecting the neutrino masses in the recent and present cosmological epoch.

V. Ultraviolet fixed point
The fixed point that is relevant for the infinite past t → −∞ ("past fixed point") corresponds to χ → 0. It is characterized by an anomalous dimension σ that appears in the scalar kinetic term. (In the language of universal critical exponents σ corresponds to −η). The approach to the fixed point corresponds to B −1 → 0, with limiting behavior of the flow equation (17) given by At the fixed point scale symmetry is exact and not spontaneously broken. With all particle masses vanishing for χ → 0 the model contains only massless modes at the UVfixed point.

Renormalized scalar field
The solution of eq. (113), contains an explicit mass scale m, in addition to the mass scale µ. For σ < 2 it can be absorbed in the definition of a renormalized field In terms of the renormalized field the effective action contains a scalar kinetic term with standard normalization For the particular case σ = 1 this yields (λ = µ 2 /m 2 ) The last term ∼ R vanishes for χ R /m → 0, such that no mass scale remains in this limit. The scale symmetry realized at the fixed point is of a non-standard type due to the non-vanishing anomalous dimension σ. While the normalized scalar field χ R scales proportional to mass, the original scalar field χ scales ∼ mass 2/(2−σ) . (For the example σ = 1 one finds a scaling of χ ∼ mass 2 ). Thus the effective action becomes invariant under the scaling The term ∼ R involves a scale symmetry violation which vanishes in the limit χ/m → 0. It characterizes a relevant parameter for the deviation from the fixed point as χ increases. For σ = 1 the term ∼ µχ 2 is invariant under the scaling (118), as easily visible in eq. (117). This situation changes for σ > 1. As an example we may consider σ = 3/2 where χ scales ∼ mass 4 and (119) Now the two last terms vanish in the limit χ/m → 0 and correspond both to relevant parameters for deviations from the fixed point. At the fixed point we are left with a free massless scalar field. This simplicity makes the existence of such a fixed point rather plausible.
We observe that for σ < 1 the coefficient e σ decreases less than ∼ m −2 for m → ∞. As a consequence the dimensionless quantity V /χ 4 R diverges for χ/m → 0. No fixed point is obtained in this case. For σ > 2 one finds that e σ increases with m, which is again not compatible with a fixed point. For the boundary case σ = 2 one finds a logarithmic dependence of χ R on χ The fixed point is now realized for χ R → −∞ where both V /χ 4 R and χ 2 R/χ 4 R vanish. We conclude that only the range 1 ≤ σ ≤ 2 is suitable for a past fixed point.
At the fixed point the term ∼ R vanishes. However, one may expect the presence of higher order invariants, as given by eq. (2). Such terms are scale invariant and therefore compatible with dilatation symmetry if the dimensionless quantities C and D are constant. (Slowly running C and D may be considered as marginal parameters for deviations from the fixed point.) In the limit χ → 0 these terms dominate the graviton propagator and the graviton-graviton scattering at nonzero momentum [58].
It will be interesting to see by an actual calculation if a fixed point with the postulated properties exists. For the moment being our model only gives an illustration of the interesting cosmological consequences of such a fixed point. If an UV-fixed point is found to exist the important task will be the understanding of small deviations from the fixed point, as encoded in the behavior of β-functions close to their zeros. This will determine the coupling ∼ χ 2 R and, for σ > 1, the term ∼ µ 2 χ 2 , as well as the flow of couplings of the standard model of particle physics. We will next address possible interesting consequences of an UV-fixed point for particle physics, in particular the gauge hierarchy problem.

Gauge hierarchy
The gauge hierarchy is related to the small value of the effective coupling ǫ H which appears in the quantum effective potential for the Higgs doubleth [3,4], For the present range of χ the function ǫ h (χ/µ) must be (almost) independent of µ and have reached a very small value ǫ h (χ/µ = M/µ) = 5·10 −33 . Besides their dependence on χ/µ the functions λ h and ǫ H also depend onĥ †h /χ 2 . This latter dependence is described by the standard model β-functions. The running of ǫ H withĥ †h /χ 2 is given by a perturbatively small anomalous dimension [59,60]. We neglect this small effect and use ǫ H ≈ 10 −32 independently ofh †h /χ 2 . (A small value of ǫ H for a Higgs field value of the order of the dynamical Planck mass χ remains small for a Higgs field value equal to the Fermi scale. This property reflects the (almost) second order character of the electroweak phase transition -the associated effective scale invariance of the non-gravitational physics protects a small value of the Higgs mass term [59][60][61][62][63][64][65][66].) We explore here if 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 the crossover for χ ≈ m.
In order to understand this issue we first consider the effective renormalized quartic coupling for the cosmon for χ ≪ m, From we extract the flow equation For 1 < σ < 2 one finds A λ < 0 and the running coupling λ R is asymptotically free in the ultraviolet. A given trajectory (model) can be specified by the value of λ R at χ/µ = 1. This is typically a rather small value, corresponding to the close vicinity to the fixed point. For larger values of χ/µ the renormalized coupling increases. We next turn to the cosmon-Higgs coupling that we define as The corresponding renormalized coupling involves χ R and the renormalized Higgs doublet h R . Using (127) Similarly, the kinetic coefficient B h of the Higgs doublet may depend on µ in the vicinity of the UV-fixed point, resulting in This relates ǫ R and ǫ H Let us now assume that ǫ R is asymptotically free in the UV, similar to λ R , This results in a flow of ǫ H according to (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 > 0, Turned around, ǫ H will then decrease for increasing χ and fixed µ. The behavior (132) is valid only for the vicinity of the UV-fixed point for χ m. For the vicinity of the standard model-fixed point, χ ≫ m, we assume that ǫ H reaches rapidly its constant fixed point value. (Formally σ ǫ ≈ 0 for 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 2 ≫ m 2 decreases by a factor This factor could explain the gauge hierarchy. For ǫ in of the order one one needs For example, for χ in = µ a value σ ǫ ≈ 5 − 6 would be sufficient for a decrease of ǫ H between χ = µ and χ = m by around 30 orders of magnitude. This would relate the smallness of the ratio Fermi scale/Planck mass and the small amplitude of primordial density fluctuations, cf. eq. , (For χ in ≪ µ smaller values of σ ǫ would be sufficient to achieve the suppression factor needed for the gauge hierarchy.) The possible emergence of a gauge hierarchy, expressed by the tiny coupling ǫ H (χ ≫ m) ≈ 10 −32 , can be viewed from different perspectives. While ǫ H should be approximately constant for χ ≫ m, nothing prevents an increase of ǫ H for χ ≪ m, such that values of the order one can be reached for small enough χ. The increase of ǫ H towards the UV-fixed point remains compatible with an asymptotically free renormalizable coupling ǫ R . For sufficiently small χ all asymptotically free renormalized couplings are very small. If the anomalous dimension |A ǫ | for the coupling ǫ R is smaller than the corresponding one for other couplings the coupling ǫ R still remains small at the crossover scale where the flow effectively stops and ǫ R roughly equals ǫ H .
The coupling ǫ H measures the distance from the electroweak phase transition which is of second order (up to small QCD-effects). This guarantees that its flow vanishes for ǫ H = 0. Such a setting generalizes to a large class of models, including grand unified models. Then ǫ H measures the distance from the hyperface in coupling constant space corresponding the phase transition. While the location of this hypersurface may be complicated in a given basis for the couplings (often associated with a "fine tuning problem") the general structure of the flow equation for ǫ H remains the same [67].
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 the gauge hierarchy [59]. The necessary large values of anomalous dimensions are often found in the gravitational contribution to the flow [12][13][14]68]. In our scenario σ has to be large in order to realize an UV-fixed point. Then also σ ǫ will typically have a large value, unless some particular cancellation occurs in eq. (131). Without an explicit computation of the µ-flow equation our discussion remains an educated guess. It clearly shows, however, that an ultraviolet fixed point with large anomalous dimension could play an important role for the gauge hierarchy problem. This also applies for a possible understanding of the value of the Higgs boson mass. If the flow of the quartic Higgs coupling λ h (χ/µ) exhibits a large positive anomalous dimension the "asymptotic safety scenario for the Higgs boson mass" is realized [69], which has led to a predicted value m h ≈ 126GeV with a few GeV uncertainties, the best value today being around 129GeV.

VI. Field relativity
Once quantum fluctuations are included on the level of the quantum effective action the corresponding field equations can be solved with arbitrary field variables. Values and correlations of physical observables are independent of the choice of fields used to describe them [43]. This exact property may be called "field relativity" [36]. Indeed, observables are expressed as functionals of fields. Again, they can be written in terms of arbitrary field variables. In general, the specific functional expressions will be changed under a change of field variables (cf. refs. [43,58] for the transformation of some quantities relevant for cosmology as temperature or proper time). We stress that only dimensionless quantities can be physical observables [43]. Different choices of field variables are called different frames. A well known example for a frame transformation is the Weyl transformation from the Jordan to the Einstein frame [70,71] that we have employed in sect. III.
In the present section we use frame transformations for several different purposes. We first show that a very large class of coupled scalar-gravity models can be brought to the form (1), with B(χ/µ) the only free function. Typically this holds if the field equations contain no more than two derivatives and the scalar potential is monotonic. A formal treatment of a large class of such models, including the ones of the Horndeski type [72], can be found in ref. [73]. Our discussion of a crossover between a past and future fixed point can therefore be carried over to a large class of models.
We have described the crossover as a "kinetial crossover" where the relevant information is encoded in the scalar kinetic term, i.e. the function B(χ/µ). Field transformations can be used to express the same physics as a "potential crossover" [74], where the information is now contained in the shape of the scalar potential V (χ/µ), while the kinetic term has a standard normalization. We also present a "primordial flat frame" for which the cosmological solution approaches flat space in the infinite past. Finally, we cast the effective action into the form of a free scalar field coupled to gravity. While the kinetic term is standard and the potential quadratic, the crossover information is now contained in a χ-dependent function multiplying the curvature scalar. Having at hand the formulation of the ultraviolet and infrared fixed points in different frames may facilitate the search for such fixed points in a genuine quantum gravity calculation.
We omit in this section higher order curvature invariants as in eq. (2). They would have to be transformed appropriately under field transformations. This section therefore deals with various expressions for the quantum effective action encoded in eq. (1).

Field transformations within Jordan frames
We will call "Jordan frames" the choice of fields for which the curvature scalar in the effective action (1) is multiplied by χ 2 . We allow for a general potential V (χ) instead of µ 2 χ 2 in eq. (1). The "Einstein frame" or "big bang frame" (30) has a constant coefficient M 2 in front of the curvature scalar. The Einstein frame is unique, up to a choice of the scalar field ϕ which may be replaced by χ or a field σ with standard normalization of the kinetic term. The Jordan frames, however, are not yet uniquely fixed, since there exist field transformations keeping the term ∼ χ 2 R invariant, while changing V (χ) and B(χ). The particular choice of fields where V (χ → ∞) = µ 2 χ 2 will be called "freeze frame". We may parametrize the Jordan frames by two dimensionless functions B(χ/µ) and Those functions contain redundant information, since they can be changed by appropriate field transformations. The quantum effective action for the most general Jordan frame takes the form (137) Using appropriate field transformations we can bring a large class of effective actions with up to two derivatives into the generic form (137). For any positive and monotonically increasing function F (χ ′ ) multiplying the graviton kinetic term −R we can choose a normalization of the scalar field F = χ 2 in order to bring the system to the Jordan frame. We can then use the residual transformation within the Jordan frame in order to obtain v = µ 2 /χ 2 such that B(χ) remains the only free function. Alternatively, we can obtain a constant scalar kinetic term at the prize of a more complicated function v.
Consider the transformation This transforms the effective action (137) to leaving the coefficient of the curvature scalar forminvariant. In terms of the variablesχ andg µν the new functionsB andχ read A pair of functions (B,ṽ) describes the same model as the pair (B, v) if the two are related by eq. (140) with a suitable choice of h(χ). The corresponding effective actions are related by a field transformation. A given model can be expressed by a whole family of Jordan frames, parametrized by h(χ).

Kinetial crossover
We can employ the field transformations (138) in order to bring a large class of potentialsṼ (χ) to the "freeze form" V = µ 2 χ 2 . Indeed, any functionṽ(χ) which decreases monotonically with limitsṽ(χ → 0) → ∞,ṽ(χ → ∞) → 0 can be transformed to v = µ 2 /χ 2 by choosing The choice of the effective action (1) is therefore rather generic, since a large family of potentials can be brought to the particular form V = µ 2 χ 2 .
As an example we may consider the models of ref. [58] With one finds whereχ is related to χ by eq. (143). For large χ/m one has χ =χ, B =B, while for χ → 0 the limiting behavior is For these models the flow equation for B exhibits two fixed points with finite values of B, e.g. 4/α 2 for µ → ∞ and 4/α 2 for µ → 0. This is the type of models investigated in refs. [4,36]. For µ → ∞ one has σ = −∂ ln B/∂ ln χ → 0. The renormalized scalar field χ R equalsχ up to a constant factor, and V /χ 4 R diverges for χ R → 0. This is not a fixed point in the sense of our previous discussion, but it could represent a possible fixed point in terms of different variables.

Potential crossover
Alternatively, we may use the transformation (140) in order to transform our models (1), (18) of a kinetial crossover to an equivalent model with a potential crossover. For this purpose we want to achieve a constantB, using for h a solution of the differential equation Once h(χ) = χ is computed in this way we can compute the associated scalar potential V (χ) =χ 4 v h −1 (χ) .
As an example we consider the effective action (1) with B(h) obeying eq. (15), For small h or large B one has the limiting behavior The solution of eq. (149) involves an integration constant This yields the potential On the other hand, for large h and small B we use B −1 = κ ln h m and therefore The solution increases faster thanχ forχ → ∞. The corresponding potential reads The full potential makes a crossover from eq. (151) for χ ≪ m to eq. (155) forχ ≫ m. The two integration constants c h andc h are related in order to ensure a smooth matching, e.g. c h ≈ exp{−κBc 2 h /8}. One may also choose a hybrid setting with a constant kinetic termB for χ → 0, while for χ → ∞ one keeps the freeze frame V = µ 2χ2 . This is achieved by choosing h(χ → ∞) =χ, while h(χ → 0) is given by eq. (150).

Primordial flat frame
Let us consider the frame where for χ → 0 the functions v andB are related by This is the condition for finding for the infinite past flat space as a solution of the field equations derived from the action (139) [4]. We can transform our crossover model (1), (3) to this "primordial flat frame" by a suitable choice of h in eq. (138). The function h(χ) has to obey a differential equation which follows from We are interested in h → 0 where B = (m/h) σ , ∂ ln B/∂ ln h = −σ. In this approximation eq. (158) is obeyed by This yields the relation between χ andχ, withm an integration constant. One infers In the primordial flat frameB vanishes forχ → 0 (which corresponds to χ → 0), in contrast to the divergence of B. The dimensionless potentialṽ =Ṽ /χ 4 diverges with an inverse power of a logarithm instead of v ∼ χ −2 . Different frames can describe the same physical situation with rather different pictures. This extends to the form of the flow equations. Forχ → 0 one has This transfers to the µ-flow equation for fixedχ andg µν (instead of fixed χ and g µν ) 5. Free scalar field coupled to gravity Another interesting frame change transforms the effective action (1) to a scalar field theory without self interactions, withχ-independentK. This is achieved by transformations that leave √ gχ 2 invariant, The transformed kinetic coefficient becomes (167) For example, one may obtainK = 0 by solving for a given B(fχ) the differential equation for f In this frame the scalar field has only gravitational interactions. The cosmological field equation expressesχ as a function ofR by the implicit equation The transformation (166) can be used in both ways. An effective action with constantK and non-trivial f (χ) can be mapped to the form (1), with In particular, for f = (χ/m)σ one finds B = (1 +σ) −2 K m χ σ + 6 1 +σ 2 For σ > 1 the asymptotic behavior B = (m/χ) σ is therefore equivalent to a positive constantK > 0 and f diverging ∼ χσ,σ = −σ/(σ − 1). The limiting case of a constant coefficient of the curvature scalar,σ = −2, corresponds to σ = 2.
On the other hand, the behavior (77) near the future fixed point for χ → ∞ can be cast into the form (165) for The fixed point corresponds to f = 1, with flow equation There are two lessons to be learned from the discussion of this section. The first concerns the generality of our description of the crossover by a varying kinetic term. The second concerns the form of the µ-flow equation underlying our approach. It depends on the choice of fields that are kept fixed as µ is varied, compare eqs. (164) with eq. (19) supplemented with µ∂ µ ln v = 2. The form of the flow equation depends o the frame. It transforms according to a variable change in a differential equation.

VII. Conclusions
We have investigated the cosmological consequences of a particle physics scenario for quantum gravity with an ultraviolet (UV) and infrared (IR) fixed point. The existence of an UV-fixed point renders quantum gravity nonperturbatively renormalizable (asymptotic safety). At this fixed point the exact scale symmetry is not spontaneously broken, such that all particles are massless. It seems possible that appropriate renormalized couplings obey asymptotic freedom. Their running is governed, however, by large (non-perturbative) anomalous dimensions. These large anomalous dimensions provide for a possible explanation of the gauge hierarchy for the electroweak symmetry breaking.
For the IR-fixed point the exact scale symmetry is spontaneously broken, resulting in massive particles and a massless dilaton. The ratio between the effective scalar potential and the fourth power of the variable Planck mass vanishes at this fixed point. In the Einstein frame this leads to an asymptotically vanishing effective cosmological constant.
Dimensionless couplings can depend only on dimensionless ratios of quantities with dimension mass. In our case this is χ/µ, where χ is the value of a scalar singlet field (cosmon) and µ the mass scale appearing in the flow equations for the running couplings. The UV-fixed point is reached for µ → ∞ or χ → 0, while the IR fixed point corresponds to µ → 0, χ → ∞. Cosmology describes a crossover from the UV-fixed point in the infinite past to the IR-fixed point in the infinite future. This is realized by a cosmological solution with χ(t → −∞) → 0, χ(t → ∞) → ∞.
The crossover between the two asymptotic fixed points is responsible for the different epochs in cosmology. We pursue models for which the crossover occurs in two distant steps, separated by a range of scales for which the flow of couplings is very slow. This range can be associated to the flow in the vicinity of an (approximate) "standard model fixed point" (SM), see Fig. 1. The range of χ and associated range in cosmological time where the SM-fixed point dominates describes the radiation and matter dominated epochs in cosmology. The UV-fixed point is responsible for the inflationary epoch, which ends at the first step of the crossover (UV→SM). The IR-fixed point will correspond to an (unknown) future scaling solution. The second step of the crossover (SM→IR) entails a transition period for the present cosmology for which dynamical dark energy (quintessence) dominates.
The cosmology of our model involves five dimensionless parameters (besides the known masses and couplings of particles of the standard model and some dark matter candidate): σ: anomalous dimension of the scalar at (or close to) the UV fixed point. It determines the spectral index n and tensor ratio r of the primordial fluctuations, cf. eq. (73). κ: coefficient of the approach to the IR-fixed point. It determines the fraction Ω ls h of early dark energy at last scattering, eq. (112). m ν0 : present average neutrino mass.
γ: present growth rate of the ratio neutrino mass/electron mass. The combinationγm ν0 determines the present fraction of dark energy Ω h (often called Ω Λ ), eq. (88).
Our model has the same number of free parameters as the ΛCDM model (e.g. n, r, A, Ω h , m ν0 ).
The overall description of cosmology by our model is simple. It describes all cosmological epochs by the dynamics of a single scalar field, the cosmon. So far our model is consistent with all observations. In the near future it is subject to interesting tests: the details of inflation (relation between n and r), early dark energy and possible consequences of large non-linear neutrino lumps. It is fascinating that a basic hypothesis about quantum gravity and the origin of mass, namely the existence of two fixed points and the necessary crossover between them becomes testable by cosmology. diverges for χ approaching the constantχ ν . This is, however, outside the validity of the approximation. Form approachingm 2 eq. (A.10) implies that the increase with χ is stopped. In the crossover region, however, the fixed point atm 2 is not yet visible. We could also multiply the r.h.s. of eq. (24) with a constant. Within the crossover region this constant can be absorbed into a redefinition ofm 1 . At this point the χ-dependence of the average neutrino mass involves two parameters,m 1 andχ ν . We will see that it corresponds to the setting of ref. [7].
The parameterm 1 is given by the ratio between the average neutrino mass and the Planck mass in earlier epochs of cosmology, before the crossover in the neutrino sector sets in. Taking in eq. (A.8) an "early value" M B−L /χ ≈ 10 −3 , as appropriate for B − L violation at some scale characteristic for grand unification, and ϕ H /χ ≈ 10 −16 , as given by the electroweak gauge hierarchy, we estimatẽ This is well compatible with our assumption that for the present cosmological epoch the neutrino masses are in the crossover region, Forγ(t 0 ) = 9 the ratio M B−L /χ has decreased at present by a factor of ten as compared to its early value.