## Cosmology Christof Wetterich

1. Early Dark Energy

Models of dynamical dark energy have been originally proposed in order to explain a small value of the “cosmological constant” or dark energy density. In these “quintessence” models dark energy decreases with time, similar to the energy density in matter or radiation. The present dark energy density is very small because the universe is very old. Typically, one finds cosmic scaling solutions (“tracker solutions”) as attractors to which many initial conditions converge. For scaling solutions, the fraction of dark energy within the total energy density of the universe is constant. This can explain dynamically why the present dark energy density in units of the Planck mass is a tiny dimensionless number ~ 10-120, similar to the present dark matter density.

For scaling solutions, the fraction in dark energy at the time of emission of the cosmic microwave background (CMB) or during the formation of cosmic structures is almost constant, say in the percent range. This presence of „Early Dark Energy” (EDE) distinguishes dynamical dark energy from a cosmological constant. For the latter the dark energy fraction at CMB-emission is tiny. Observation of Early Dark Energy could therefore distinguish between dynamical dark energy and a cosmological constant.

After earlier work on the influence of EDE on the precise location of the peaks in the CMB-anisotropies and the modification of the growth rate of cosmic structures, we have investigated the effects of EDE on non-linear cosmic structures as clusters of galaxies. Our first investigations (201) have been followed by very substantial work by other groups, including sophisticated numerical N-body simulations. Constraints on the amount of EDE for various cosmological epochs have been derived in (202), (209), (285).

It can be shown (294) that a very large class of modified gravity theories is equivalent to quintessence models with a coupling of the cosmon – the scalar field of quintessence – to matter. Such models of “coupled quintessence” have been investigated by many groups and often exhibit Early Dark Energy.

2. Growing neutrino quintessence

While cosmological scaling solutions can explain why the dark energy density is of the same order as the dark matter density – and therefore tiny in units of the Planck mass – they fail to account for the observed acceleration of the cosmic expansion in the present epoch. An accelerated expansion follows if the dark energy density is almost constant. In models of dynamical dark energy or quintessence the time evolution of the cosmon or quintessence scalar field should be much slower in the present epoch as compared to the earlier epochs where the scaling solution is valid. Some ingredient has to end the scaling behavior and effectively stop the time evolution of the cosmon. This “stop” has to happen a couple of billion years ago, say around redshift image067.png. This raises the “why now problem”, i.e. the question why such a stop occurs in the present cosmological epoch, and not much earlier or much later.

Neutrinos have been found (214), (215) to be prime candidates for an explanation of the “why now problem”. Due to the coupling between neutrinos and the cosmon field they trigger an effective stop of the cosmon evolution as soon as they become non-relativistic. Given the known range for neutrino masses this stop indeed occurs at redshift image067.png. This correlates the present dark energy density with the present neutrino mass and solves the “why now problem”.

More precisely, the ratio of neutrino mass over electron mass is assumed to depend on the value of the cosmon field and increases as the cosmological value of the cosmon increases. In such models of “growing neutrino quintessence” (GNQ) the behavior of cosmology is different for the epochs where neutrinos are relativistic or non-relativistic. For relativistic neutrinos their mass plays no role, and the cosmon-neutrino coupling is ineffective. In contrast, once neutrinos are non-relativistic the neutrino energy density is dominated by the mass. The cosmon-dependence of the neutrino mass induces an additional term in the field equations for the cosmon which is proportional to the neutrino energy density, multiplied by the cosmon-neutrino coupling β. This term counteracts the effect of the gradient of the cosmon potential, leading to the effective stop of its evolution.

Cosmological models of this type exhibit two steps for the cosmon evolution. In early cosmology the cosmon follows a scaling solution, explaining why the dark energy density is of the same order as the dark matter density. Once neutrinos are non-relativistic the cosmology looks very similar to a cosmological constant. It is then hard to distinguish observationally the cosmological background evolution from a model with a cosmological constant. In this respect the main distinctive feature is Early Dark Energy.

The precise timing of the crossover between the scaling solution and dark energy domination is determined by the growth rate γ of the neutrino mass. The number density of the cosmic neutrinos being known from their decoupling in the early cosmic plasma, the present dark energy density is found proportional to image068.pngν, with image069.pngν the present average neutrino mass. The observed value of the present dark energy density is obtained for image070.pngν image071.png 6 eV, a very successful relation for image069.pngν in the sub-eV range allowed by present observational bounds and γ of the order one.

For image072.png of the order one, the present bounds on Early Dark Energy imply a substantial cosmon-neutrino coupling image073.png. In turn, this coupling mediates an additional attractive force between neutrinos, stronger than the gravitational attraction by a factor image073.png2. Typical values are image073.png2 image074.png, implying a substantial enhancement of the attraction between neutrinos as compared to the gravitational attraction for dark matter or atoms. While the cosmon mediated attractive interaction for neutrinos is much too small to be observed in any laboratory experiment or star explosion, it has important consequences for the cosmological evolution of the cosmic neutrino background.

Typically, the time scale for the growth of anisotropies in the cosmic neutrino background is by a factor 1/image073.png2 shorter than the one for dark matter structures – the latter being set by gravity. The neutrino anisotropies have been tiny at the moment when neutrinos became non-relativistic, similar to anisotropies in the photon background. Nevertheless, the rapid subsequent growth makes the relative neutrino fluctuations to become of the order one, and therefore non-linear, already at redshift image075.png (223). Subsequently, large scale neutrino structures form on length scales up to hundreds of Mpc, only somewhat smaller than the size of the observable universe. These structures may collapse to stable neutrino lumps, or periodically be dissolved and reformed, drawn by oscillations in the mass of neutrinos.

Stable neutrino lumps have first been investigated in (217). Their presence could be visible as large spots in the cosmic microwave background (243) or by enhanced peculiar velocities (246). We have attempted first analytic and numerical quantitative estimates of the dynamical formation of the large scale neutrino structures (247) and their influence on the observed power spectrum (262). Due to strong non-linearities an analytic understanding is only partially possible. In particular, neutrinos bound in lumps typically have smaller mass than the unbound ones. This non-linearity leads to strong effects of “backreaction” (264), by which the formation of cosmic neutrino structures modifies the overall cosmological evolution, as compared to a situation where the neutrino background is taken homogeneous. The gravitational potential generated by neutrino structures modifies the propagation of light and could therefore leave observable imprints in the CMB anisotropies (260). A quantitative estimate needs the size of the neutrino-induced gravitational potential and its time evolution.
For a quantitative understanding of the non-linear dynamics of neutrino lumps a numerical N-body code has been developed (267) (274) (282) (297) (303). It follows the trajectories of neutrinos with variable mass, including situations where neutrinos are relativistic. This is important because neutrinos become accelerated so fast during the lump formation that they turn again relativistic, even if they have already been non-relativistic before (267). In addition to the neutrino fluid and dark matter the code has to provide the spatial distribution of the scalar cosmon field at every time step. This is crucial since the local neutrino mass depends on the local value of the scalar field, inducing additional acceleration or deacceleration due to the locally varying mass. Furthermore, backreaction is taken into account by determining the actual average energy density and pressure from the given distribution of matter, neutrinos and the cosmon, rather than using a homogeneous background model. Several ingredients developed in this context, as the deviation from geodesic motion due to varying mass or relativistic effects, find application in other cosmological models, as modified gravity.

The overall outcome of our numerical simulations has revealed two classes of dynamics that depend on the details of the growing neutrino quintessence model. For a field independent cosmon-neutrino coupling image073.png the evolution leads to a fluid of neutrino lumps that do not dissolve anymore and can be described approximately by a neutrino-lump fluid (282). The backreaction becomes so large that the effective stop of the cosmon evolution is substantially weakened, showing now substantial deviations from a cosmological constant. It seems difficult to obtain compatibility with observation for this type of model (303).

Growing neutrino quintessence motivated by a present crossover period (see 3.)) lead to a substantial change of neutrino mass (in the Einstein frame) only during a certain cosmic epoch. In such models, the coupling image076.png is an increasing function of the cosmon field (215). In this type of models the oscillations in the cosmic neutrino mass are so strong that the large scale neutrino structures form and dissolve periodically (297). Neutrino induced gravitational potentials do not grow very large, and backreaction effects are small. The effect of large scale cosmic neutrino structures may be too small to be observable. The overall cosmology resembles since redshift image067.png quite closely the one for a cosmological constant. For certain ranges in the neutrino mass a distinction from a cosmological constant may not be possible in the near future. Inbetween the two extreme classes of models one suspects a substantial range of models for which the structures in the cosmic neutrino background would lead to observable signals. The observable effects of such large scale cosmic anisotropy are under present investigation.

3. Fixed points in quantum gravity and cosmology – the role of scale symmetry

Quantum gravity is a non-perturbatively renormalizable quantum field theory if an ultraviolet (UV) fixed point exists. This concept of asymptotic safety has been supported in recent times by functional renormalization group studies in a multitude of different truncations. We have also proposed the presence of an infrared (IR) fixed point in quantum gravity. The presence of fixed points in quantum gravity may have deep consequences for cosmology.

We have explored cosmological scenarios where the infinite past of the universe approaches an UV-fixed point, while the infinite future is characterized by the IR-fixed point. At every fixed point scale invariance is an exact symmetry of the quantum effective action and the cosmological field equations derived from it. Since the flow generators (β-functions) for running couplings and consequently their zeros, which correspond to the fixed points, originate from quantum fluctuations, the dilatation invariance at the fixed points may be called quantum scale symmetry.

In our approach the approximate scale symmetry close to the UV-fixed point is reflected by the approximate scale invariance of the spectrum of primordial cosmic fluctuations. At the infrared fixed point scale symmetry becomes an exact symmetry of the effective action, but is spontaneously broken by the cosmological expectation value of a scalar field χ. The spectrum of particle masses therefore contains massive particles besides the massless graviton and photon. Spontaneously broken scale symmetry is reflected by the presence of an exactly massless Goldstone boson- the dilaton. In present cosmology we are close to, but still away from the IR-fixed point. Scale symmetry is only approximative, resulting in an almost massless scalar field (pseudo Goldstone boson). This extremely light field is the cosmon. It is responsible for the present dynamical dark energy. In the infinite future the mass of the cosmon will go to zero.

Inbetween the UV-and IR-fixed points a crossover has to occur. Typically, couplings run fast in the crossover regions, and scale symmetry is explicitly violated. If the UV-fixed point has more than one relevant (or marginal) direction the crossover proceeds in several stages that are often separated by many orders of magnitude in scale. In our scenario the crossover regions in the flow of couplings are reflected by transitions periods in cosmology. One is the end of inflation, another one the present onset of dark energy domination.

Scale symmetry of the quantum effective action is most easily realized by the presence of a scalar field χ besides the metric. The fixed Planck mass M in the Einstein-Hilbert gravitational action is replaced by the dynamical field χ, such that the modified gravitational action is scale invariant if the metric and χ are scaled simultaneously. Any non-zero cosmological value of χ results in spontaneous scale symmetry breaking and induces a non-vanishing variable “Newton constant” image077.png. After the first crossover at the end of inflation scale symmetry is (almost) exact in the sector of particles of the standard model of particle physics. Scale symmetry implies that the masses of electrons or protons, as well as other particles and binding energies are all exactly proportional to χ, and dimensionless couplings are independent of χ. This explains why despite the cosmological evolution of χ all ratios of particle masses and dimensionless “fundamental constants” are (almost) constant, as required by observation. The field χ approaches zero at the UV-fixed point and diverges at the IR-fixed point. It plays the role of the inflaton in the early inflationary stage of the universe, and the cosmon in the present dark energy dominated epoch.

The connection between running couplings in quantum gravity and the cosmological evolution proceeds by a simple mechanism. The flow of couplings is described by their dependence of a “renormalization scale” µ. On the other hand, the relevant couplings are dimensionless. They must therefore be dimensionless functions of the ratio χ/µ. This connects the µ-dependence of couplings in the effective action to their dependence on the dynamical scalar field χ. The UV-fixed point image078.png at fixed χ can also be seen as the limit χimage079.png0 at fixed µ. Similarly, the IR fixed point image080.png0 corresponds to χimage081.png. The cosmological field equations are derived from the quantum effective action by variation with respect to the metric and with respect to χ. Solving these field equations one finds that χ increases from zero to infinity as conformal time image082.png increases from minus infinity to plus infinity. Physical time t, as measured by the number of oscillations of a “clock”, is well approximated by conformal time and extends from minus infinity to plus infinity. The time evolution of χ therefore maps the dependence of couplings on χ to their dependence on cosmic time t. By the chain image080.pngχ, χimage083.png the flow of couplings in quantum gravity is mapped to the cosmological evolution.

An overall description of the role of UV- and IR-fixed points for cosmology can be found in (298). This has led to fully consistent overall models of cosmology involving no more free parameters than the models with a cosmological constant and inflation. These models of “variable gravity” are described in 6.).

Investigations of possible consequences of the UV-fixed point in the matter sector have led to a prediction (250) of the Higgs boson mass mH=126 GeV, with a few GeV uncertainties. The mass found later by LHC experiments is within this range.

First, not yet finally conclusive functional renormalization group studies of an IR-fixed point (288) have seen such fixed points in simple truncations. Interestingly, all candidate fixed points found have the property that a quartic scalar interaction image084.pngχ4 is absent. The potential for the scalar field V(χ) increases with χ slower than χ4. Scale symmetry violation near the fixed point may lead to a quadratic increase for χimage081.png, V=µ2χ2, or a constant V=µ4.

The observable quantity related to a cosmological constant or, more generally to dynamical dark energy, is dimensionless and given by the ratio of V divided by the fourth power of the effective Planck mass. (For dynamical dark energy there are additional contributions from the time derivative of χ.) In our picture the Planck mass is given by χ. The relevant ratio V/χ4 therefore decreases image084.pngµ2/χ2 or ~ µ4χ4 for χimage081.png. The effective “cosmological constant” goes to zero asymptotically. It is possible to use a different “Einstein frame” by a field transformation of the metric. In the Einstein frame the Planck mass and particle masses take constant values. Appropriate dimensionless quantities are independent of the choice of frame. In the Einstein frame the potential V goes to zero for χimage081.png. The role of scale symmetry and its small violation (scale anomaly) for the absence of a cosmological constant in the Einstein frame, and the naturalness of an exponential potential for the cosmon in this frame have been discussed extensively in (222).

4. Variation of fundamental couplings

Exact quantum scale symmetry at a fixed point implies that dimensionless couplings and mass ratios are independent of χ. They remain constant even if χ and therefore all masses increase. Scale symmetry can provide a profound explanation why no variation of fundamental couplings has been observed so far.

Still, in the present cosmological epoch the universe is away from the IR-fixed point that will only be reached in the infinite future. One therefore expects small violations of scale symmetry. They are reflected by tiny variations of dimensionless “fundamental constants” as the fine structure constant or the ratio between electron and proton mass. “Fundamental couplings” show a small dependence on χ and therefore on cosmic time. The size of these variations is difficult to estimate. It depends on anomalous dimensions for the vicinity of a fixed point, which determine how close a coupling is to its asymptotic constant value. Observations of tiny variations of fundamental couplings can be interpreted as signs for dynamical dark energy. No such variation is expected for a cosmological constant.

We have performed a comprehensive analysis (212) how a possible time variation of couplings would affect the element abundancies resulting from nucleosynthesis. Several different fundamental couplings influence the outcome of nucleosynthesis. We have therefore divided the analysis and the corresponding numerical code into several steps. The first step investigates how element abundancies vary as a result of the variation of several phenomenological parameters as nuclear binding energies and cross sections or masses of atoms. In a second step the variation of the phenomenological parameters is related to changes in the “fundamental couplings” of the standard model of particle physics, as the fine structure constant, or the electron mass, Fermi scale or Planck mass in units of the nucleon mass. A third step may relate variations of different couplings of the standard model, for example in the context of grand unification. From the observed primordial element abundancies, we have derived bounds on how much fundamental constants can have varied from nucleosynthesis until now.

A similar analysis has been performed for a possible variation of couplings in other cosmological epochs. This concerns, in particular, redshift zimage071.png1-2, for which an observation of varying couplings has been claimed, or the present time for which high precision experiments are performed and have to be compared to existing bounds as from the Oklo natural reactor. A time variation of fundamental couplings is probed by sophisticated precision experiments. In (237) we compare the relative sensitivity of experiments and recent time observational bounds from Oklo or meteorites, considering only few independent variations of fundamental couplings.

A relation between the time variation of couplings at different epochs, say nucleosynthesis, CMB-emission, observed quasars at redshift zimage071.png2 or present or historical variations on earth, need knowledge about the cosmological evolution of the cosmon field χ. It is the variation of χ that induces the variation of the couplings. For epochs of a slow variation of χ, as the present epoch in models of growing neutrino quintessence, only a very small variation of couplings is expected. We have computed the variation of couplings in various models of dynamical dark energy (229), (232).

5. Dilation symmetry in higher dimensions – possible consequences for dark energy and dark matter.

At an infrared fixed point dimensionless couplings take constant values. A crucial ingredient is the fixed point value of the quartic cosmon self-interaction appearing in a term image085.pngχ4 in the cosmon potential. If the IR-fixed point value of image085.png, that is reached for χ/image086.png, e.g. image087.png = image085.png (χ/image086.png), differs from zero one would observe an effective cosmological constant with value image088.pngin units of the Planck mass M. For realistic cosmology a non-zero value image087.png would have to be a tiny number image087.png image071.png 10-120. This does not seem natural. A solution of the cosmological constant problem by dynamical dark energy associated to an approach to an infrared fixed point requires image087.png = 0. The approach to this fixed point is then governed by an anomalous dimension A, image087.png ~ (image089.pngA, and realistic dynamical dark energy models can be obtained.

The precise reason why a possible fixed point occurs at image087.png = 0 is not established so far. First functional renormalization group investigations of possible infrared fixed points have only found (288) candidates if image087.png = 0, but the reason is not fully understood. It may be noted in this context that a value image087.png image090.png 0 is consistent with scale symmetry, but not with conformal symmetry. Another possibility could relate image087.png = 0 to particular properties of dilatation symmetry in more than four dimensions. Higher dimensional theories offer additional adjustment mechanisms for obtaining a zero value of image085.png.

Such additional adjustment mechanisms are associated with warped geometries. The geometry of “internal space” spanned by the additional space dimensions must be such that internal space is not directly observable – typically because relevant length scales are much smaller that the wavelengths used in experiments and can therefore not be resolved. If internal space does not exhibit maximal isometry the characterization of its geometry involves functions (as the warping function and others), and not only a few couplings or parameters. Whole functions, and not only a finite number of couplings, can participate in a possible adjustment mechanism.

It was observed long ago that the most general solution of the gravitational field equations for warped spaces admit free integration constants, and that the effective four dimensional cosmological constant after dimensional reduction is among them. In the language of four-dimensional variable gravity the value of image085.png for static solutions, i.e. image087.png, would be a free integration constant. This issue has been studied in much details for warped branes with codimension two (203). The long standing issue of these free integration constants has been partially clarified (225) by noting that particular values are selected if one requires an extremum of the action not only locally in internal space, but also averaged over internal space.

Still, the warped solutions of higher dimensional theories have particular properties if the action from which the field equation is derived shows scale invariance. In the presence of higher dimensional dilatation symmetry all stable static solutions have image085.png = 0, while possible solutions with image087.png image090.png 0 are all unstable (227) (248). If there exists a static solution to which the universe converges in the infinite future, and if this corresponds to a fixed point and therefore realizes asymptotically scale symmetry, one concludes that image087.png must be zero. The geometry of internal space has to perform a dynamical adjustment leading to image087.png = 0 (257).

The degree of freedom that describes in an effective four-dimensional language this adjustment would be the cosmon. There could be more than one effective scalar field involved in the adjustment mechanism. Simple models describe then two light scalar fields participating in the present cosmic dynamics. The lightest one is the cosmon, with effective mass of the order of the Hubble parameter. The second one may have a substantially larger mass, for example somewhat larger than the inverse galactic size. Such a field, called “bolon”, would be a candidate for dark matter (261). Since there is a common geometric origen of the cosmon and bolon both fields are typically coupled, realizing coupled dark energy.

Interestingly, these simple models show a correlation between the onset and amplitude of scalar oscillations and the mass of the bolon. For a realistic amount of dark matter the inverse mass turns out to be of subgalactic size. A possible interesting observational consequence is a cutoff in the dark matter power spectrum at subgalactic scales. We have investigated (296) the detailed time history of bolon dark matter and its effect on the power spectrum. A cutoff on subgalactic scales may be detectable by observation. Unfortunately, the bolon as a dark matter candidate would not be observable in laboratory experiments since its interactions are only of gravitational strength.

6. Universe without expansion – variable gravity

In a world with variable particle masses the universe does not need to expand. The observation of redshift for distant galaxies only tells us that the dimensionless ratio of galaxy distance divided by the size of atoms is increasing. For fixed particle masses, and therefore a fixed size of atoms, this necessitates an expansion of the intergalactic distances – the expanding universe. For increasing particle masses, and therefore a shrinking size of atoms, no geometrical expansion is needed. The intergalactic distances may even decrease, leading to the geometric picture of a shrinking universe. The observed redshift of distant galaxies is then simply explained by the fact that the electron mass was smaller in the past than at present, leading to an emission of light at smaller frequencies and therefore shifted towards the red. Since light needs time to travel from distant galaxies to us, the light of these galaxies was emitted in the past, such that spectral lines are red-shifted as compared to the ones observed in present laboratories.

A shrinking universe was colder in the past as compared to the present. Instead of a hot big bang we now deal with a “freeze picture” where the temperature of the universe was extremely low in the past, increasing only slowly. This picture is fully in line with the observation of the microwave background radiation or nucleosynthesis. Indeed, what matters for the physical plasma, that existed before atoms were forming, is the ratio temperature over particle mass. If particle masses increase proportional to the value of a scalar field image091.png, and temperature only increases at a slower rate, say image092.png, the ratio temperature over mass was higher in the past than at present. The physical properties of the plasma in the freeze picture are the same as for the expanding universe, leading to nucleosynthesis or the emission of the cosmic microwave background.

What is needed to turn such a freeze picture of cosmology into a model where observable quantities can be computed from field equations is a setting where particle masses can increase in the course of the cosmic evolution. This is easily realized by a generalization of the Higgs mechanism. If all particle masses are proportional to the value of a scalar field image091.png, and the solution of the field equation for image091.png is such that it increases with time, the particle masses increase with time. Consistency with observational bounds implies that also the Planck mass is proportional to image091.png and varies with time. Such models of “variable gravity” are consistent with all observational bounds and realize the freeze picture of cosmology.

We have found simple models of variable gravity (290) that realize a “universe without expansion” (287) (293), for which the geometric distances in the universe shrink during the epochs of radiation and matter domination. During these two epochs the geometry is a de Sitter universe with a decreasing scale factor. The model can also describe an early stage of inflation. Exact scale invariance would imply an exact proportionality of all quantities with dimension mass and χ. This includes the Fermi scale as well as the strong interaction scale, and therefore all particle masses, binding energies or cross sections. For such a fixed point dimensionless couplings are independent of χ. In the standard model sector of particle physics this is assumed to hold for all renormalizable couplings since the crossover that has ended inflation. A small residual scale violation in the potential for χ can lead to scaling solutions for dynamical dark energy.

Viable models of present dynamical dark energy can be found if one assumes that in the “beyond standard model (BSM) sector” a new crossover occurs approximately in the present cosmological epoch. A crossover implies varying mass ratios and couplings. It affects cosmology by a variation of the non-renormalizable couplings of the standard model, that show a direct connection to the BSM sector. Prime candidates are neutrino masses. A variation of the ratio between neutrino mass and electron mass realizes growing neutrino quintessence.

There exists an exact map between the freeze picture of cosmology and an equivalent “big bang picture”. This map involves a non-linear field transformation of the metric, which is multiplied by a function of χ (Weyl scaling). “Field relativity” states that observables, that can be defined by the quantum effective action and its functional derivatives, remain invariant under field transformations. Field relativity has far reaching conceptual implications since geometry looses its absolute meaning.

7. Absence of big bang singularity – eternal universe

The freeze picture of the universe has helped to understand the nature of the singularity that occurs in many models of inflation within the big bang picture. This singularity turns out to be the consequence of an inappropriate choice of fields, rather than being a physical singularity. Such “field singularities” can be avoided by a more suitable choice of fields that are used to describe observations. It is conceptually similar to the “coordinate singularity” at the north pole that arises in certain maps of the earth. Other maps cover the north pole without any particular problems.

For the fields used in the freeze picture, the cosmic solution that is equivalent to usual inflation models can be continued to the infinite past in time. This concerns not only some “coordinate time”, but typical definitions of physical time as realized by counting the ticks of a clock – say the number of oscillations of some quantum field. For such solutions the universe is eternal – it extends from the infinite past to the infinity future. It can be verified that physical time also extends to minus infinity in the big bang picture of inflation. Clocks tick an infinite number of times as the big bang is approached. The equivalence of physical time in the freeze and big bang pictures confirms field relativity.

The origin of the apparent singularity in the big bang picture can be understood easily from the point of view of variable gravity and the freeze picture. In the infinite past χ goes to zero and all particle masses vanish. This is typical for an ultraviolet fixed point for which scale symmetry is not spontaneously broken. Vanishing particle masses pose no particular conceptual problems in quantum field theory. The explicit cosmological solutions show no singularity. One should recall, however, that proper time ceases to be a suitable measure of time for massless particles, and resort to a more suitable definition of physical time.

In the Einstein frame the particle masses are fixed. For a situation where “physical particle masses” go to zero as χ image093.png0, the map from the freeze frame to the Einstein frame must become singular for χ image093.png0. This field singularity in the map to the Einstein frame is the origin of the big bang singularity. In short, definitions of inverse time and other quantities with dimension of mass in units of particle masses become unsuitable if the physical particle masses go to zero. This is precisely what happens in the big bang picture as well. What counts for the concept of a physical particle mass is the dimensionless ratio mass / momentum. For example, this decides if a particle is relativistic or not. At the big bang singularity in the Einstein frame the particle masses remain constant, but all momenta diverge. Thus physical particle masses go indeed to zero – all particles become relativistic.

We have compared (295) the description of several models in the big bang or freeze frame. This has established “frame invariant” observables that are the same in both pictures and represent physical observables. This investigation has confirmed the absence of a physical singularity at the big bang. It is even possible to choose fields for which the cosmological solutions are such that flat Minkowski space is approached in the infinite past (290)(293)(298). For these pictures the absence of singularities is most apparent.

8. Cosmon inflation

A form of homogeneous dark energy is assumed to dominate the universe at present and during the very early stage of inflationary cosmology. These two epochs show several similarities. They are, however, separated by a huge difference in scale. For inflation, the energy density stored in the scalar potential at the time when observable fluctuations leave the horizon is a factor 10108 larger than the present dark energy density.

Cosmon inflation (286) describes inflation and present dynamical dark energy by the same scalar field χ- the cosmon. This can be achieved within variable gravity by a simple scalar potential V = image094.png2 χ2. The observable quantity characterizing homogeneous dark energy is V divided by the fourth power of the variable Planck mass χ. The huge decrease in the ratio V/ χ4 = image094.png2/ χ2 during the long duration of the intermediate radiation and matter dominated epochs accounts for the huge difference in scale for the two dark energy dominated epochs.

Inflation needs an end, and present dark energy domination needs a beginning. Both events can be related to cosmic crossover periods, resulting from crossover phenomena in the properties of the quantum effective action. If one normalizes χ such that the variable Planck mass equals χ, the coefficient B-6 in front of the kinetic term involves a dimensionless coupling, the kinetical B. (In variable gravity stability requires B >0.) For a large value of B the evolution of the cosmon is slow and such a period describes inflation. For small values of B cosmology follows in the presence of radiation or matter a scaling solution with almost constant dark energy fraction. It is sufficient that B is a running coupling that reaches large values for small χ/image095.png. Inflation ends once B becomes of the order one. The end of inflation is therefore related to the crossover in the flow of B from large to small values as χ/image095.png increases. The onset of present dark energy domination may be related to the crossover in a different sector, as for growing neutrino quintessence.
We have investigated different models of inflation, as characterized by different functions for the kinetical B or potential V. The computation of amplitude and spectrum of primordial scalar and tensor fluctuations has revealed large classes of viable inflationary scenarios. A minimal scenario assumes an increase of B image071.png (m/χ)Ϭimage096.pngfor χimage097.pngwith anomalous dimension ϭ characteristic for the approach to an ultraviolet fixed point (298). In this scenario both the scalar spectral index n and the tensor amplitude r can be computed in terms of ϭ and are therefore related. In this minimal scenario the tensor amplitude r comes out rather large, r image098.png0.1. Such a value could be detected or excluded in the near future by polarization measurements of the CMB.

9. Primordial cosmic fluctuations

Primordial cosmic fluctuations constitute the key observable signal from inflation. The standard description investigates small “classical” deviations from homogeneity as solutions of the linearized Einstein equations. These homogeneous differential equations cannot fix the amplitude of the fluctuations, however. The latter is provided by assuming as initial condition the fluctuations of a free quantum field in an assumed vacuum for the curved background geometry. While very successful in practice, this description has a few conceptual shortcomings. The transition from the quantum to the classical description is ad hoc. The criteria for the selection of the vacuum are not very obvious. It is left open how the quantum fluctuations are “created” and how different initial conditions may influence the observed spectrum. Furthermore, a map between different frames (different field choices for the metric) is complicated, since the rules are not known how to map quantum operators and their commutation relations between different frames.

For these reasons we have developed (302) a different formal approach for the computation of primordial cosmic fluctuations. The relevant quantity is the two-point correlation function G from which the observable amplitude and spectrum of the fluctuations can be extracted. We employ the standard basic definition of a Greens function or correlation function by an inhomogeneous differential equation DG = 1. The differential operator D is given by the second functional derivate of the quantum effective action. The unit on the right hand side typically involves image099.png–distributions in space-time or momentum space. This formulation starts from an exact equation – assumptions and approximations only concern the precise form of the effective action. The basic quantity G makes no distinction between classical or quantum fluctuations. No operators appear – the normalization of the amplitude results from the inhomogeneous term image084.png 1. No particular assumptions for “vacua” are needed. The form of a differential equation clearly defines the fluctuation problem as an initial value problem. The mapping between different frames can be done by field transformations in the quantum effective action and poses no particular problem.

For suitable initial conditions one recovers the results that arise from the standard assumption about the vacuum within the usual treatment – namely the Bunch-Davies vacuum for de Sitter space. Other initial conditions lead to other “vacua” that have been proposed in the literature. The Bunch-Davies vacuum seems particularly natural. It is realized whenever the correlation function for the high momentum modes has approached the Lorentz invariant correlation function for flat space during the early stages of inflation.

The investigation if correlation functions approach the “scaling form” corresponding to the Bunch-Davies vacuum, and how long such an approach takes, can now be done by methods of non-equilibrium quantum field theory. At the moment there is no settlement of the question if equilibration occurs towards an attractive scaling solution, with the associated effective loss of memory of initial information. The equilibration time is unknown. We have performed an investigation of scalar (302) and tensor (304) fluctuations for geometries characteristic for inflation. For an approximation of the effective action constaining only terms with up to two derivations we have found the general solution of the initial value problem for rotation symmetric correlation functions. Beyond previously known pure quantum states it includes also mixed states. On the level of this approximation no effective loss of information of initial conditions occurs.

This raises the problem to which extent the observations of the CMB-anisotropies can reveal information about the inflaton potential, or if the information contained in the observed spectrum dominantly or partly concerns information about unknown “initial conditions” at the “beginning of inflation” (301). Only if memory of initial conditions is effectively lost, reliable information about the inflaton potential can be gained. Although the amazing statement remains true that we can gain observational information about the period very close to the big bang, it is not clear if we “see” the beginning of inflation, or rather an epoch towards its end when the wavelength of observational fluctuations goes out of the horizon. In case of effective equilibration this issue depends on the equilibration time and the time how long inflation has lasted before horizon crossing.

Our investigation of primordial fluctuations in variable gravity (304) clearly demonstrates the equivalence of different frames. The relevant fluctuation quantities as the primordial power spectrum turn out to be the same in all frames. We discuss models with increasing particle mass for which geometry is close to Minkowski space during inflation. An almost scale invariant spectrum is now due to the variation of the gravitational “constant”. This finding demonstrates clearly that a geometric horizon is not necessary in order to protect the primordial information from being erased in the thermal equilibrium state during radiation or matter domination.