Emergent scale symmetry: Connecting inflation and dark energy
Quantum gravity computations suggest the existence of an ultraviolet and an infrared fixed point where quantum scale invariance emerges as an exact symmetry. We discuss a particular variable gravity model for the crossover between these fixed points which can naturally account for inflation and dark energy, using a single scalar field. In the Einstein-frame formulation the potential can be expressed in terms of Lambert functions, interpolating between a power-law inflationary potential and a mixed-quintessence potential. For two natural heating scenarios, the transition between inflation and radiation domination proceeds through a “graceful reheating” stage. The radiation temperature significantly exceeds the temperature of big bang nucleosynthesis. For this type of model, the observable consequences of the heating process can be summarized in a single parameter, the heating efficiency. Our quantitative analysis of compatibility with cosmological observations reveals the existence of realistic models able to describe the whole history of the Universe using only a single metric and scalar field and involving just a small number of order one parameters.
A dynamical scalar field with a sufficiently flat potential and at most tiny couplings to ordinary matter is often advocated as a promising alternative to the cosmological constant Wetterich:1987fm (); Ratra:1987rm (). This idea, usually named quintessence, can partly be viewed as a late-time implementation of the successful inflationary paradigm. Since inflation and dark energy share many essential properties, it is natural to seek for a unification of these two mechanisms into a common framework Peebles:1998qn (); Spokoiny:1993kt (); Brax:2005uf (). In this paper, we postulate that inflation and dark energy are intimately related to an underlying symmetry: scale invariance.
When dealing with scale invariance one can take two different perspectives: i) assume that scale invariance remains an exact symmetry even when quantum corrections are taken into account Shaposhnikov:2008xi () or ii) assume that scale invariance is broken by quantum effects but will be approximately realized close to fixed points Wetterich:1987fm (). Cosmological models based on the first line of reasoning and their associated phenomenology can be found in Refs. Fujii:1974bq ()-Kannike:2016wuy (). Cosmological models of the second type resulting in a dilatation anomaly that vanishes asymptotically in the infinite future led to the first proposal of dynamical dark energy or quintessence Wetterich:1987fm (); Wetterich:1994bg ().
In this work we will adopt the second point of view. In particular, we will assume that scale invariance is generically broken by the conformal anomaly but it reemerges as an exact quantum symmetry in the early- and late-time evolution of the Universe. The resurgence of the symmetry can be related to the presence of ultraviolet (UV) and infrared (IR) fixed points in the renormalization group flow. In the vicinity of these points, any information about the mass scales in the theory is lost Wetterich:2014gaa (). This idea can be easily implemented in a variable gravity scenario Wetterich:2014gaa (); Wetterich:2013jsa (); Wetterich:2013aca ().
In this paper we present the complete cosmological history for a particular crossover variable gravity model with a singlet scalar field. In the scaling frame the field is coupled nonminimally to gravity and to the Standard Model, supplemented by some unspecified dark matter candidate and potentially by heavy particles as in grand unification. The model contains no tiny or huge dimensionless quantities put in by hand. The four parameters appearing in the effective action are all of order one. The first three describe the approach to the UV and IR fixed points in the scalar sector and the position on the crossover trajectory. The last parameter describes the present growth rate of neutrino masses, which is associated to the coupling between the scalar field and neutrinos. For early cosmology, the net effect of the interactions between the scalar field and the Standard Model particles (and possible sectors beyond that) can be summarized in a heating efficiency . These few parameters are sufficient for a quantitative account of the history of the Universe from inflation to the present accelerated expansion era. Our simple model seems so far compatible with cosmological observations. Neither tiny nor fine tuned parameters are introduced to explain the small value of the present dark energy density, which is rather a consequence of the long age of the Universe in Planck units.
The comparison of our model with cosmological observations is performed in the Einstein frame with a canonical kinetic term for the scalar field. This allows us to find explicit analytic solutions and facilitates the comparison with other quintessential inflation models in the literature, see for instance Refs. Peebles:1998qn (); Spokoiny:1993kt (); Brax:2005uf () for well-known examples and Refs. Hossain:2014xha (); Agarwal:2017wxo (); Ahmad:2017itq (); Geng:2017mic () for recent discussions. We follow here the general approach in which the inflationary epoch is followed by a transition to a scaling or tracker solution of which the long duration is responsible for the tiny value of the present dark energy density. The end of this scaling era is triggered by neutrinos with growing masses that become nonrelativistic in the recent cosmological history. This general scenario, originally proposed in Refs. Wetterich:2013aca (); Wetterich:2013wza (), has been recently followed by several groups Hossain:2014xha (); Agarwal:2017wxo (); Ahmad:2017itq (); Geng:2017mic (). The explicit Einstein-frame formulation presented in this paper allows us to replace arguments for approximate solutions by exact analytical results, which substantially extends the range of validity of the scenario in parameter space.
Beyond the explicit and convenient solutions in the Einstein frame, our investigation contains several new results. We propose for the heating or entropy production preceding the radiation dominated epoch a general mechanism that is neither gravitational particle production nor instant preheating. Only the latter two mechanisms have been previously discussed within models of quintessential inflation. The mechanism presented in this paper is based on the general framework for particle production in the presence of time-varying fields, but adapted to the situation in which the potential does not have a minimum. The absence of a minimum is required for the transition from inflation to a tracker solution and typical for a variable gravity framework containing a single crossover at early times. For this scenario all features relevant for observations can be summarized into a single parameter – the heating efficiency . The duration of the kination epoch between the end of inflation and the onset of the radiation dominated epoch can be rather short, leading to a high heating temperature. We find that the (almost massless) cosmon excitations generated during the heating stage do not significantly contribute to the effective number of neutrino species at big bang nucleosynthesis.
This paper is organized as follows. In Section 2, we present the effective action of the model in a scaling frame where the Planck mass is given by a scalar field. We describe the properties of the UV and IR fixed points responsible for the early- and late-time acceleration of the Universe. In Section 3, we reformulate the variable gravity scenario into the more common, although completely equivalent, Einstein frame. This formulation is used in the following sections to study the cosmological implications of the model. Section 4 contains the details of inflation. We show that the UV fixed point gives rise to a power-law Einstein-frame potential and derive the associated inflationary observables. The spectral tilt and the tensor-to-scalar ratio are shown to be related and to depend only on the UV fixed-point anomalous dimension. The initial stages of the postinflationary dynamics are discussed in Section 5. The crossover to the IR fixed point translates into the appearance of a field region where the Einstein-frame potential becomes steep. This triggers the onset of a kinetic domination regime. The kinetic regime must be limited in time for the model to be cosmologically viable. In particular, part of the energy density of the inflaton field must be transmitted to the Standard Model particles, which must become the dominant energy component before big bang nucleosynthesis (BBN). In Section 6, we discuss two natural heating mechanisms and determine the associated radiation temperature (“reheating” temperature). We argue that a total decay of the inflaton field is not possible, and is neither necessary nor even preferable. The evolution after heating and the onset of the dark energy dominated era are discussed in Section 7. Section 9 contains our conclusions. Appendix A summarizes several properties of Lambert functions that are useful for the derivation of the analytic solutions presented in this paper. Appendixes B and C contain details of our heating scenario and of the creation of cosmon excitations during this period.
2 Variable gravity scenario
The variable gravity scenario is usually formulated in a scaling frame in which not only the Planck scale, but also the dimensionless couplings and masses of elementary particles are allowed to depend on the expectation value of a scalar field . We consider here a simple real scalar which plays simultaneously the role of the inflaton, the cosmon, or the dilaton. The effective Lagrangian density for the graviscalar sector of the theory reads Wetterich:2014gaa (); Wetterich:2013jsa (); Wetterich:2013aca ()
where the tilde denotes quantities in the scaling frame and we have suppressed Lorentz indices. The implicit contractions in this paper should be understood in terms of the metric associated with the frame under consideration.
The cosmon field in Eq. (1) defines the effective variable Planck mass. We will see that for the cosmological solutions of the field equations derived from the action (1) it increases with time, with and . The only fixed scale not proportional to the cosmon field is the scale , which is associated to the scale or dilatation anomaly. The value of has no intrinsic meaning and can be used to set the mass scales. We will take
We have chosen to normalize the scalar field by its coupling to curvature in the scaling frame, i.e. by the first term in the right hand side of Eq. (1). With this normalization, the scalar kinetic term has typically a nonstandard normalization, as reflected by the dimensionless function . In order to have a well-defined kinetic term during the whole cosmological evolution, we will require the function to be a positive function of . For and constant the associated action is scale invariant, while for the action is also conformally invariant and the cosmon field no longer propagates.
For the matter and radiation sectors we take the Standard Model of particle physics with possible extensions including dark matter. We assume that at large the values of all the (renormalizable) dimensionless couplings in the Standard Model become independent of , as required by scale symmetry. In practice, this implies that the Fermi scale and the confinement scale of strong interactions are proportional to . The masses and binding energies of all elementary particles are then proportional to the dilaton expectation value, while cross sections scale as . In consequence, our setting is compatible with the equivalence principle tests and the severe bounds on the variation of fundamental constants Uzan:2010pm ().
A recent quantum gravity computation based on functional renormalization has indeed found for variable gravity a quadratic increase of the scalar potential for large Wetterich:2017ixo (). A strong enhancement of the effect of long-distance graviton fluctuations avoids a potential instability of the graviton propagator that would arise for a potential increasing faster than . More generally, large classes of effective actions containing no more than two derivatives can be brought to the form (1) by appropriate nonlinear field redefinitions Wetterich:2014gaa (). For example, this concerns potentials of the form .
A given model is specified by a choice of . For successful quintessential inflation one needs large during the inflationary epoch and small after the end of inflation. Large ensures slow-roll dynamics during inflation, while inflation ends once gets small. In this paper, we will concentrate on a particular scenario where satisfies the flow equation
This equation contains an infrared fixed point , approached for with a quadratic term
The ultraviolet fixed point for
is characterized by an anomalous dimension .
No quantum gravity computation for the flow of is available so far. Eq. (3) should be therefore understood as an educated guess, or an assumption, on the exact quantum gravity dynamics. As suggested by the first investigations in Ref. Henz:2016aoh (), we assume the renormalization flow of quantum gravity to admit both a UV and an IR fixed point. The enhanced conformal symmetry for implies that the -function for vanishes for . If the -function in the IR limit is analytic in around , i.e. , the assumption of a vanishing infrared anomalous dimension motivates the limit (4). A simple way of achieving large in the UV limit is an anomalous dimension of the scalar wave function renormalization, leading to the limit (5). The precise interpolation between the UV and IR fixed points in Eq. (3) is not important for the observable consequences of the model. A reason for the selection of the particular crossover in Eq. (3) is its simplicity. The scalar-gravity sector contains only three order one parameters: two constants and and an integration constant selecting a particular trajectory in the flow. The resulting tensor-to-scalar ratio of primordial perturbations turns out to be comparatively large, Wetterich:2014gaa () (see also Ref. Hossain:2014xha ()). Smaller values of can be obtained by modifying the behavior of at small , for example by assuming a fixed point of the flow at some large but finite Wetterich:2013wza (), instead of the limit (5).
Since the main points of this paper will not be affected by the details of the function , we will take advantage of the simplicity of Eq. (3) for finding explicit solutions. Indeed, Eq. (3) can be easily integrated to obtain
or equivalently (cf. Eq. (125))
with the Lambert function refLambert () and
a crossover scale related to the integration constant via dimensional transmutation.
3 Einstein-frame formulation
Most of the literature on inflation and on dynamical dark energy employs a canonically normalized scalar field in the Einstein frame. In order to permit an easy access and comparison of models for a wider community, the investigations and results of the present paper will be performed in this setting. The transformation of our variable gravity scenario to the Einstein frame will be done in two steps. The first realizes the Einstein frame with a fixed Planck mass and a noncanonically normalized scalar field. The second step proceeds to a canonical normalization of the scalar kinetic term.
Performing a conformal transformation , with dimensionless scalar potential
and reduced Planck mass GeV, we obtain
One could also take . As we will see below, the constant will completely disappear after canonically normalizing the scalar kinetic term.
Due to the positive definite choice of in Eq. (1), the Einstein-frame Lagrangian (10) is ghost free. In this basis, the cosmon potential decays exponentially to zero Wetterich:1987fm (); Wetterich:1994bg () and the dynamical information is encoded in the kinetial Wetterich:2013jsa (); Wetterich:2013wza (), see also Ref. Dimopoulos:2017zvq ().
The kinetic term can be made canonical by performing an additional field redefinition
The relation between and is given by (see also Fig. 1)
denotes a transition field value lying between the UV and IR fixed points. Indeed Eq. (15) implies
Integrating this expression we get the identity
In the canonical basis, the action takes the standard form
All dynamical information is now encoded in the effective potential , as given by Eqs. (15) and (16). The second term in Eq. (16) contains a linear piece in . The potential in Eq. (15) is therefore nonsymmetric for arbitrary . For one gets and Eq. (15) becomes a power-law (chaotic) potential
On the other hand, for one has and can be approximated by a mixed-quintessence potential
We recall that the ratio in Eq. (25) is related to the integration constant determining the particular trajectory in the flow (cf. Eq. (8)). Order one values of translate naturally into values of that are exponentially smaller than one, . This provides for a natural explanation of the small amplitude of the primordial fluctuations.
4 Inflationary era
We can now proceed to discuss the observable consequences of our model by using the standard methods developed for a canonically normalized scalar field in the Einstein frame. If correctly defined and computed, the observable predictions cannot depend on the particular frame under consideration nor on the precise scalar-field normalization. This is indeed verified by the following independent computations. These computations put our variable gravity framework in direct contact with the known properties of inflationary potentials and dynamical dark energy scenarios existing in the literature.
The approximate power-law form of the potential at allows for inflation with the usual chaotic initial conditions. The Einstein-frame equation of motion for the cosmon field in a flat Friedmann-Lemaître-Robertson-Walker Universe
is given by
with dots denoting derivatives with respect to the coordinate time and . The Universe undergoes a phase of accelerated expansion if
The evolution of the acceleration parameter (29) can be determined by numerically solving Eq. (28) together with the Friedmann equations and standard slow-roll initial conditions. Depending on the value of , the end of inflation for the inflationary potentials (15) and (24) can take place at slightly different field values (the smaller the transition scale , the smaller the difference). This change translates into a small variation in the number of -folds for the values of and we are interested in. Having this in mind, we will estimate the inflationary observables using the simple power-law approximation (24).
Following the standard procedure, we obtain the following expressions for the spectral tilt and the tensor-to-scalar ratio
the number of -folds between the horizon crossing of the relevant fluctuations () and the end of inflation (). These Einstein-frame results improve the estimates in Ref. Wetterich:2014gaa () by properly identifying the end of inflation with . 111The estimates in Ref. Wetterich:2014gaa () replace the denominators in Eq. (30) by , due to a small change in the precise definition of the end of inflation. While in the present work, the offset of inflation is defined to occur at , Ref. Wetterich:2014gaa () takes . The comparison of (30) with the latest cosmic microwave background (CMB) results is shown in Fig. 3. For an anomalous dimension , the inflationary predictions lay within the Planck/BICEP2 contour Ade:2015lrj (); Ade:2015xua ().
The amplitude of scalar perturbations
and the associated trajectory in the flow. For and , one obtains . Our results agree with Ref. Wetterich:2014gaa ().
Once we have determined the Einstein-frame inflationary dynamics, we can always reinterpret our results in terms of the original variable gravity formulation. In particular, it is interesting to compare the value of the cosmon field during inflation with the scales and . Combining Eqs. (9) and (32) we get
Evaluating the result at horizon crossing we get
with the tensor-to-scalar ratio in Eq. (30). The scale is indeed associated to the end of inflation
A simple overall picture of inflation arises. The inflationary phase corresponds to the vicinity of the UV fixed point for . Close to a fixed point approximate scale symmetry is manifestly realized. This approximate symmetry is the origin of the almost scale invariant primordial fluctuation spectrum. For one observes the crossover from the vicinity of the UV fixed point to the vicinity of the IR fixed point. Scale invariance is substantially violated in this crossover region. This violation triggers the end of inflation. The scale is an integration constant of an almost logarithmic flow. Small values of arise therefore naturally, similarly to the small ratio of the confinement scale in quantum chromodynamics as compared to some “unification scale”. This provides for a small fluctuation amplitude
without the necessity for tuning.
5 Kinetic dominated era
After inflation, the inflaton rolls down into the steep potential (26), leading to a substantial decrease of the potential energy density. The evolution of the cosmon field becomes dominated by its kinetic energy and the heating of the Universe sets in. In this section, we consider the initial epoch in which the energy density into radiation is still small as compared to the energy density of the cosmon. Such a period is usually referred as kination or deflation Spokoiny:1993kt (). During this epoch, we can continue to use the cosmon-field equation (28). Neglecting the potential energy density in the first approximation, the equation with admits a solution
with and the value of the field and its derivative at the onset of the kinetic era at . During this regime the cosmon energy density scales as . This behavior is reflected by the cosmon equation of state
In Fig. 4, we show the numerical solution of the field equations for as a function of the number of -folds . The equation of state evolves rapidly toward after the end of inflation.
For a realistic cosmology, the kinetic domination regime has to end before big bang nucleosynthesis. For this era to begin, the energy density of the cosmon field must be (dominantly) transmitted to the Standard Model degrees of freedom. In a nonoscillatory model like the one under consideration, a total decay of the inflaton field is not expected since is not a solution of the equations of motion in the absence of a minimum. Also, highly effective processes such as parametric resonance cannot take place in a nonoscillatory model.
Although an incomplete inflaton decay would constitute a serious drawback for most inflationary models, it does not in the variable gravity scenario. To understand this, let us assume that a given heating mechanism is able to produce a partial depletion of the cosmon condensate by the creation of relativistic particles. Even if the energy density of this component is initially very small, it will inevitably dominate the energy budget at later times. Indeed, during the kinetic dominated regime the energy density of the created particles scales as , while that of the cosmon field evolves as . The rapid decrease of the cosmon energy density will inevitably give rise to a late-time domination of the radiation component.
The radiation temperature at which the energy density of the created particles equals that of the cosmon () can be defined as
with the effective number of relativistic degrees of freedom at that temperature. The quantity should be interpreted as the typical energy scale for the onset of radiation domination. It coincides with the heating temperature (usually called the reheating temperature) in the fast thermalization limit.
For a simplified scenario, we may assume particle production to take place instantaneously at the onset of the kinetic regime. This motivates the introduction of a heating efficiency, defined in this limit as
We will later extend the definition of to smoother transitions. For the types of heating or entropy production mechanisms considered in this paper the parameter is sufficient for a quantitative description of the cosmological history.
For instant particle production the heating efficiency can be easily related to the radiation temperature by taking into account that
the temperature of the created particles at the onset of kinetic domination.
The longer is the kinetic regime, the smaller is the radiation temperature . The heating efficiency (43) must be large enough to avoid conflicts with BBN. For the power-law inflationary potential (24), the Hubble rate at the end of inflation/onset of kinetic domination is of order GeV (cf. Table 1), with a slight dependence on the precise value of . Taking this into account, Eq. (44) becomes
The inflationary dynamics not only excites cosmon fluctuations but also generates primordial gravitational waves (GW). In the postinflationary era, the amplitude of GW with superhorizon wavelengths remains constant until it reenters the horizon. When that happens, the logarithmic GW spectrum scales as Sahni:1990tx ()
with the effective equation of state. For a radiation dominated expansion, the GW spectrum remains flat. However, for a kinetic dominated regime, the spectrum becomes blue tilted and may eventually dominate the total energy budget.
Nucleosynthesis constraints set an integral bound on the GW density fraction at BBN, namely Maggiore:1999vm ()
with and and the momenta associated respectively to the horizon scale at the end of inflation and at BBN. The dominant contribution to this integral comes from momenta that left the horizon before the end of inflation and reentered during kinetic domination. For these modes () Giovannini:1999bh () (see also Refs.Sahni:1990tx (); Giovannini:1998bp ()),
the dimensionless amplitude of gravitational waves. The present radiation content in critical units is given by
takes into account the variation on the number of massless degrees of freedom between thermalization and decoupling Giovannini:1999bh (). For the Standard Model content (, ), we have .
The ratio in this expression can be easily related to Eq. (43) by taking into account that
Using these results, the integral bound on the GW density fraction at BBN can be translated into a lower bound on the heating efficiency
For the typical values of in Table 1 (and assuming ), we get
Using Eq. (46), this translates into a lower bound on the radiation temperature
Among the different heating mechanisms that have been proposed in the literature (see for instance Refs. Peebles:1998qn (); Hossain:2014xha (); Chun:2009yu (); Tashiro:2003qp (); Sami:2004xk (); Feng:2002nb (); Dimopoulos:2002hm (); Liddle:2003zw (); Sami:2003my ()), there are two that can be naturally realized in a variable gravity framework: heating via gravitational interactions and heating via matter couplings involving strong adiabaticity violations. In the following, we will estimate the contribution of these heating scenarios to the heating efficiency (43) and the associated radiation temperature (44).
6.1 Heating via gravitational interactions
The simplest and most minimalistic heating mechanism is particle creation via gravitational interactions Spokoiny:1993kt (); Ford:1986sy (); Damour:1995pd (). Scalar fields nonconformally coupled to the metric tensor are inevitably produced in an expanding background,222Note however that this mechanism does not apply to gauge bosons and chiral fermions since their evolution equations in a conformally flat geometry as Friedmann-Robertson-Walker are invariant under Weyl rescalings. provided that they are light enough as compared to the Hubble rate. In our scenario, the scalar sector contains the Higgs doublet and the cosmon, but it can also include additional scalars as those appearing in extensions of the Standard Model such as grand unification.
During a Hubble time, the gravitationally induced variation of the (relativistic) scalar energy density is associated to an effective (Hawking) temperature . This effect has to compete with the dilution due to the expansion of the Universe, . During kinetic domination and . In consequence, gravitational particle production is dominated by times close to the onset of the kinetic epoch, while later particle production becomes negligible. On the other hand, during the inflationary epoch is almost constant and , with the numbers of -folds. Particle creation during the early stages of inflation is therefore exponentially diluted and can be also neglected. We conclude that sizable entropy production due to gravitational interactions concerns only the epoch immediately after inflation.
As seen in Fig. 4, the kinetic dominated era starts soon after the end of inflation. The energy scale of the relativistic scalars created at the onset of this regime is of order
resulting in a radiation temperature
with the effective number of (scalar) relativistic degrees of freedom at the transition from inflation to the kinetic epoch.
If the created scalar particles are allowed to interact after production via nongravitational interactions,333Note that this does not apply to gravitational waves, which cannot thermalize below the Planck scale Giovannini:1998bp (). they will rapidly generate a thermalized plasma that should contain, at least, the Standard Model degrees of freedom. In that case, the radiation temperature can be associated to the heating temperature. The effects of partial thermalization can be incorporated into a modification of the efficiency parameter .
Note that, although is typically above the BBN temperature MeV, it is not high enough to satisfy the bound (56) for a moderate number of scalar fields. Indeed, combining Eqs. (55) and (59) (and assuming ) we get
Thus, even for efficiency, a large number of scalar fields is required in order to satisfy the GW constraints.
Independently of the plausibility of Eq. (61), gravitational particle production should not be considered a completely satisfactory heating mechanism. As argued in Ref. Felder:1999pv (), the presence of light fields during inflation could give rise to unwanted effects, such as the generation of secondary inflationary periods or the production of large isocurvature perturbations. As we will show in the next section, these problems, together with the inefficiency of gravitational particle production, can be easily solved in the presence of direct couplings between the cosmon field and matter.
6.2 Heating via matter interactions
After Weyl rescaling the coefficients in the quadratic part of the effective action for matter fields generically depend on (see Ref. Wetterich:1987fk () for the Higgs doublet). For a scalar field , this dependence can be parametrized as
The -dependence of the effective action induces particle production if the coefficients or change substantially with time. Rapid variations of these functions are expected to occur during the crossover, where the dimensionless couplings and mass ratios of matter fields must evolve from their UV fixed-point values to those associated to the IR fixed point.444In the Einstein frame, the matter fields must eventually decouple from the cosmon to avoid violations of the equivalence principle Uzan:2010pm (); Wetterich:2003qb (). This is realized if an IR fixed point is approached.
To understand how a change in the effective couplings translates into particle production, let us consider the Einstein-frame equation of motion for the field
For the field equation for contains derivative interactions. For the sake of simplicity, we will neglect this coupling in the following considerations and set .555Derivative interactions give rise to similar particle production effects, see for instance Ref. Lachapelle:2008sy (). For a homogeneous cosmon field (), the mode equation in Fourier space reads
The friction term in this expression can be eliminated by performing a field redefinition . Doing this, we get a time-dependent harmonic oscillator equation
The term is responsible for the gravitational particle production discussed in Section (6.2). In the presence of direct couplings between the inflaton and matter fields this term is expected to be subdominant and it will be neglected in what follows.
The solutions of the mode equation (66) could be used to compute the propagator for the -field in the time-dependent background , along the lines of Ref. Wetterich:2015gya (). From this, particle creation can be directly extracted. We will follow here the more conventional approach based on the operator formalism.
Let us describe the solutions of the mode equation (66) in terms of positive- and negative-frequency adiabatic solutions , namely
The time dependence of the functions and has to ensure that the mode equation (66) is obeyed. We will require and to satisfy the differential equations
These conditions induce the following evolution equations for and
In particular, the Wronskian condition
is preserved in time. The condition (77) arises in the operator formalism from the commutation relations of creation and annihilation operators for free fields. Extracting the propagator as the inverse of the second functional derivative of the effective action, this condition is induced by the inhomogeneous term in the propagator equation Wetterich:2015gya ().
The occupation number of particles at time can be identified with
The number and energy density of the created particles is therefore given by
one infers the relation
This, together with the condition , translates for the occupation numbers to
Vacuum initial conditions correspond to and and therefore to . In terms of these initial conditions become
The production term in the right-hand side is indeed proportional to . It has to compete with the Hubble damping in the left-hand side.
In the absence of derivative interactions (i.e. for in Eq. (62)), the cosmon field couples to the matter field only through the effective mass function . In a natural and phenomenologically successful scenario, this function should satisfy the following criteria:
It should be large enough during inflation to retain the single-field inflationary picture and avoid the generation of large isocurvature perturbations.
It should rapidly vary at the end of inflation () to heat the Universe via violations of the adiabaticity condition .
It should eventually become independent of if the field is the Higgs doublet. This reflects scale symmetry in the Standard Model sector as required by the bounds on the variation of the ratio of the Fermi scale over the Planck scale since nucleosynthesis.
We assume here that the crossover from the UV to the IR fixed point, which is reflected in the change of the cosmon kinetic term and associated to the end of inflation, leaves also its traces in the matter sector. In the field range , the couplings of to , and correspondingly to , are therefore expected to undergo significant changes. This provides for a natural scenario where can change rapidly from large to small values at the end of inflation. The conditions i)-iii) can be viewed as the imprint of the crossover in the matter sector.
6.3 Workout example
A possible parametrization of satisfying the above requirements is with
and , , and are taken to be positive constants. The shape of mimics the form of the cosmon potential (15), with , and the amplitude left free. The detailed structure of this parametrization is chosen for illustration purposes only. Alternative choices sharing the features described in i), ii) and iii) could be used without modifying the conclusions below.
The behavior of Eq. (85) for different values of is shown in Fig. 5. It describes the evolution from a UV fixed point666In the far UV (), this corresponds in the scaling-frame to a flow equation (87) with .
approached for with anomalous dimension , to an IR fixed point where . The constant encodes the location of the transition. The smaller the value of , the longer the effective coupling stays in the vicinity of the UV fixed point.
For large values of , we can make use of Eq. (35) to relate this parameter to the crossover scale signaling the end of inflation
Values of larger than correspond to values of smaller than the crossover scale . For , the transition in occurs within a short period before the end of inflation. In this region, the size of mainly determines the sharpness of the crossover, with small leading to a more abrupt transition. We could introduce and additional parameter for the timing of the transition, e.g. by replacing in Eq. (85) by . All our models can account for a gauge hierarchy if with for () and for ().
The occupation numbers and the energy density of created particles for a given set of parameters can be computed by numerically solving the mode equation (66) with vacuum initial conditions. Let us consider for concreteness an anomalous dimension in an inflationary model with and . For this choice of parameters, the interaction Lagrangian during inflation () contains a quartic term with . To retain the predictions of single-field inflation, we will require the effective mass of the scalar field during inflation to be larger than the mass of the cosmon. In the leading order approximation (24) the squared cosmon mass reads
This expression vanishes for and is generically suppressed by the small factor , cf. Eqs. (25) and (33). Unless is tiny, the effective mass of the field during inflation will be significantly larger than the cosmon mass. For our practical example we choose , while keeping as a free parameter. For we take , which translates into an asymptotic mass of order GeV) at . Smaller values of , as those required if is the Higgs doublet, will not change our discussion.
The numerical results for different values of are summarized in Fig. 6 and Table 2. All cases are evaluated at the onset of the kinetic domination regime. At that time, and . In agreement with the analytical estimates presented in Appendix B, smaller choices of translate into more significant particle production.