Quantum corrections to inflation: the importance of RG-running and choosing the optimal RG-scale
Matti Herranen, Andreas Hohenegger, Asgeir Osland, Anders Tranberg
QQuantum corrections to inflation: the importance of RG-running and choosing theoptimal RG-scale
Matti Herranen a,b,c , ∗ Andreas Hohenegger d , † Asgeir Osland d , ‡ and Anders Tranberg d § We demonstrate the importance of correctly implementing RG-running and choosing the RG-scale when calculating quantum corrections to inflaton dynamics. We show that such correctionsare negligible for single-field inflation, in the sense of not altering the viable region in the n s − r plane, when imposing Planck constraints on A s . Surprisingly, this also applies, in a nontrivial way,for an inflaton coupled to additional spectator degrees of freedom. The result relies on choosing therenormalisation scale (pseudo-)optimally, thereby avoiding unphysical large logarithmic correctionsto the Friedmann equations and large running of the couplings. We find that the viable range ofparameters of the potential is altered relative to the classical limit, and we find an upper limit of g (cid:39) − on the value of the inflaton-spectator portal coupling still allowing for inflation. And anupper limit of g (cid:39) − for inflation to correctly reproduce the scalar amplitude of fluctuations A s . Keywords: cosmological perturbations, inflation, cosmology, quantum field theory in curved space-time
I. INTRODUCTION
Observations of the Cosmic Microwave Background in-dicate a period of accelerated expansion in the early Uni-verse, known as inflation. The predominant theoreticalrealisation of such an epoch is through the potential en-ergy of a slowly rolling scalar field, the inflaton. Thechoice of the potential function V ( φ ) largely determinesthe observable spectrum of scalar and tensor metric fluc-tuations.Traditionally, model building and predictions rely onthe classical dynamics of a homogeneous field in an ex-panding background metric, and in many cases the sim-plification of expressing evolution equations in terms of aset of slow-roll parameters, equations that may then betruncated at some order in powers of these parameters.But ultimately, this homogeneous field must be identi-fied as the quantum expectation value of a quantum fieldˆ φ (1-point function, mean field, condensate), subject toa complete quantum field theoretical treatment. Then V ( φ ) must be identified as the quantum effective poten-tial .Various choices of V ( φ ) in model building must there-fore ideally be connected to an underlying quantum the-ory, restricting the form that such a potential may take.As an example, it is well known that an interacting the-ory at tree level generates a tower of interactions at looplevel, and that the couplings run with the renormalisa-tion scale. In particular, non-minimal coupling to gravityis generated away from the conformal limit, and field self-interactions generically generate logarithmic dependenceon the field. ∗ matti.h.herranen@jyu.fi † [email protected] ‡ [email protected] § [email protected] This is not the low-energy effective potential, obtained from inte-grating out heavier degrees of freedom, but the quantum effectivepotential from integrating out all fluctuations and is a functionof the mean field only.
Quantum corrections are often argued to be ”small” insome sense, and can be controlled by powers of the cou-plings. For single-field inflation, the self-interactions aretypically small to provide the necessary flatness of thepotential during the inflationary epoch. On the otherhand, the renormalisation scale is a priori arbitrary, andit is tempting to choose it to augment the effects of run-ning [4, 5]. This misses the point that the renormalisa-tion scale is a parametrisation of a perturbative diagramtruncation. A large dependence on this scale is a signalthat perturbation theory is unreliable, and so to trust thecomputation, it is no longer arbitrary, but subject to acriterion of ”small dependence”.Quantum corrections to inflation has been consideredseveral times before (see for instance [2–21] for a broadrange of approaches), with some works reporting signif-icant quantum effects, some negligible effect. Becausetaking all aspects of the calculation into account is quiteinvolved, often focus is put on a few such aspects, whileneglecting or ignoring others (typical examples are theinclusion or omission of RG-running, curvature effects,IR effects and resummations, scalar metric fluctuations).Often, a slow-roll treatment is introduced, which requiresthat the effective potential in the inflation equation ofmotion is the same as in the Friedmann equation. Thisis in general not the case (see for instance [19]).In this work, we consider two models of inflation: onewith a single self-interacting field. And one where thissingle field is coupled to a spectator field. We computethe 1-loop effective potential and 1-loop (or 2-loop) RG-running, include curvature effects, but neglect IR-effects.The size of the quantum contributions, throughout theevolution, depends manifestly on the RG-scale. As weexplain, there is no truely optimal choice of this scale forthe second model, but our results suggest one that is pre-ferred, by far, over keeping it constant. By performingcomprehensive parameter scans using a modified versionof MultiModeCode [22] , we investigate for both models Which does not rely on a slow-roll approximation for solving the a r X i v : . [ h e p - ph ] O c t whether, when using a (pseudo-)optimal, time-dependentRG-scale, quantum corrections may still be important.We find that for a single inflaton field, the smallness ofthe self-interaction makes the corrections negligible. But,when coupled to other fields, the size of that coupling isa-priori unconstrained (within reason), potentially lead-ing to a significant modification in the basic CMB pre-dictions. However, we will see that while this is true forthe individual trajectory, the overall region of n s − r pro-duced by these inflation models are largely unchangedwhen including quantum corrections.As for many other works on this subject, we willalso rely on the ”semi-classical” approximation to quan-tum corrections in curved space-time, where the scalarmetric fluctuations are ignored. The quantum scalarfield(s) evolve in a classical Friedmann-Robertson-Walker(FRW) background. Including metric fluctuations is dis-cussed in some details in [21], where we argue that forlarge-field inflation as considered in the present work, thesemi-classical approach can be expected to be reliable.The structure of the paper is as follows: In section II weintroduce the single-field model, slow-roll and quantumcorrections as well as the RG-running and show how tochoose the RG-scale. In section III this model is subjectto a numerical sampling and computation of the CMBspectrum including quantum corrections. We analyse themagnitude of the corrections from their projection ontothe n s − r plane. In section IV we extend the single-field model with a ”portal” coupling to a second scalarin its vacuum, and describe again the quantum correc-tions, RG-running and one convenient choice of RG-scale.Section V is then a numerical sampling of the resultingmodel. We again consider the impact on the observation-ally allowed region in an n s − r -diagram, and analyse towhat extent our convenient choice of RG-scale is optimal.Section VI is a technical description of the numerical pro-cedure and modifications to MultiModeCode, an existingoff-the-shelf package underlying our code. We concludein section VII. II. ONE SELF-INTERACTING, MASSIVEFIELD, NON-MINIMALLY COUPLED TOGRAVITY
We consider a single-field inflation model, with the ac-tion S = (cid:90) d n x √− g (1) × (cid:34) M f ( φ ) R + 12 ∂ µ φ ∂ µ φ − m φ − λ φ (cid:35) , field dynamics. and where we choose the specific case f ( φ ) = 1 − ξφ M . (2)This is the simplest and most common non-minimal cou-pling between gravity and matter. A. Classical slow-roll
For a non-minimally coupled theory, the usual slow-rollparameters (cid:15) = − ˙ HH , (cid:15) = − η = ¨ φH ˙ φ , (3)must be supplemented by (see for instance [23–28]) (cid:15) = − ξφ ˙ φHM f ( φ ) , (cid:15) = 6 ξ (cid:0) ξ − (cid:1) φ ˙ φH (cid:16) M + 6 ξ (cid:0) ξ − (cid:1) φ (cid:17) . (4)When all of these are (cid:28)
1, the Universe is inflating, andthe inflaton field is slow-rolling. It then makes sense towrite down and expand the equations of motion and ob-servables in powers of (cid:15) , , , . In a flat FRW background,the classical field equation of motion then reads3 H ˙ φ (cid:18) (cid:15) (cid:19) = − (cid:0) m + ξR (cid:1) φ + 16 λφ , (5)with the scalar curvature R = 12 H + 6 ˙ H = 12 H (1 − (cid:15) / φ . The Friedmann equations followfrom variation of the action with respect to a generalmetric g µν , and subsequently specialising to FRW space.We find3 M H f ( φ )(1 + 2 (cid:15) ) == 12 ˙ φ + 12 m φ + λ φ , (6)3 M H f ( φ ) (cid:18) (cid:15) − − (cid:15) (1 + 12 (cid:15) ) (cid:19) == 12 (1 − ξ ) ˙ φ − m φ − λ φ . (7)In addition, we define the quantity δ = V (cid:48)(cid:48) H = m + ξR + λφ H , (8)which, for a 1-field model of inflation, is related to leadingorder in slow-roll as δ (cid:39) (cid:15) − (cid:15) ). In the slow-roll limit(i.e. neglecting (cid:15) , , , relative to constant of order 1), thefield- and Friedmann equations become3 H ˙ φ = − (cid:20)(cid:0) m + 12 ξH (cid:1) φ + 16 λφ (cid:21) , (9)3 M H f ( φ ) = 12 m φ + λ φ . (10)The standard procedure is then to define the end of infla-tion as (cid:15) = 1, back-track the evolution of φ and H a timecorresponding to N e-folds, and compute the basic CMBobservables at this ”horizon crossing epoch”. These arefor a non-minimally coupled model given by A s = 14 π H ˙ φ , (11) n s − − (cid:15) − (cid:15) + 2 (cid:15) − (cid:15) , (12) r = 16( (cid:15) + (cid:15) ) , (13)for the pivot-scale amplitude A s , scalar spectral index n s and scalar-to-tensor ratio of amplitudes r , respectively.Results based on solving (5, 6, 7) we will refer to as clas-sical , to distinguish them from the quantum correctedevolution, we will introduce in the following. Similarly,(9, 10) will denote classical slow-roll . The expressions ofthe central observables, eqs. (11, 12, 13) follow from theevolution of the quantum modes of the metric fluctua-tions. For the 1-loop, 1PI quantum treatment we per-form in the following, they are unaltered as expressionsin slow-roll parameters. They may of course take on dif-ferent values in case the quantum corrected dynamicsproduces trajectories in ( φ, ˙ φ, H, ˙ H ) different from theclassical ones. B. Quantum corrections
In the semi-classical approach we may straightfor-wardly compute the 1-loop effective potential in an FRWbackground (see for instance [16]), we find in the MS-scheme V eff = 12 m ( µ ) φ ( µ ) + 12 ξ ( µ ) Rφ ( µ ) (14)+ 124 λ ( µ ) φ ( µ ) + 164 π M ( φ ) (cid:20) log | M ( φ ) | µ − (cid:21) . We have introduced the mass (squared) M ( φ ), whichappears in the conformal-time field mode equation. Itreads M ( φ ) = m ( µ ) + 12 λ ( µ ) φ ( µ ) + (cid:18) ξ ( µ ) − (cid:19) R. (15)The couplings m ( µ ), λ ( µ ), ξ ( µ ) and the field φ ( µ ) arenow running with the renormalisation scale µ , relative to It is not the second derivative of the potential. some reference scale µ . In the following, we will suppressthe explicit µ in our notation, but keeping in mind thatwhenever quantum corrections are included, all the pa-rameters run with this scale. In the classical approxima-tion they do not. By variation of the effective potential,we find the equation of motion for the mean field¨ φ + 3 H ˙ φ + (cid:0) m + ξR (cid:1) φ + 16 λφ + λφM ( φ )32 π (cid:18) log | M ( φ ) | µ − (cid:19) = 0 . (16)This is an equation for the evolution of the renormalisedfield φ and is expressed only in terms of RG-improvedcouplings.The Friedmann equations do not follow from varia-tion of the effective action (14), but from variation of theclassical action and, again in the semi-classical approach,computing the appropriate quantum expectation values[21]. The result is 3 M H = T C + T Q ,a M H (cid:18) (cid:15) − (cid:19) = T Cii + T Qii , (17)where T Cµν denotes the classical energy-momentum ten-sor, and the quantum correction is T Qµν = − g µν H π (cid:18) log | M ( φ ) | µ − (cid:19) (18) × (cid:2) δ − δ(cid:15) − δ − (cid:15) + 12 ξ (2 − δ + (cid:15) − δ(cid:15) ) (cid:3) . Since the object δ is not a-priori a slow-roll parameter,for illustration we have included contributions to lead-ing order in (cid:15) and all orders in δ . φ will from now onbe taken to be responsible for the CMB fluctuations, inwhich case it is a light field in the sense that δ can beconsidered leading order in slow-roll. Hence in the fol-lowing, we will discard three of the eight terms in (18) ashigher order in slow-roll ( δ , − δ(cid:15) , 12 ξδ(cid:15) ). C. RG-running and the choice of RG-scale
The RG-improved couplings λ , m , ξ follow from solv-ing the 2-loop RG equations [4, 5]1 µ dλdµ = 3 λ (4 π ) (cid:18) − λ (4 π ) (cid:19) , (19)1 µ dm dµ = m λ (4 π ) (cid:18) − λ (4 π ) (cid:19) , (20)1 µ dξdµ = (cid:18) ξ − (cid:19) λ (4 π ) (cid:18) − λ (4 π ) (cid:19) + λ π ) . (21)The solution is not easily written in closed form, and al-though we include it in our numerical integration, someof the analysis below will be performed at 1-loop for il-lustration. The renormalised field φ is also a function ofthe scale through the anomalous dimension, γ = λ π ) , (22)by means of wave-function renormalization: φ ( µ ) = φ c e − (cid:82) µµ dµ (cid:48) γ ( µ (cid:48) ) µ (cid:48) . (23)Since the action and dynamics will be expressed entirelyin terms of the renormalised field, the ”classical” field φ c will not enter explicitly.It remains to choose the renormalisation scale µ , andthe reference scale µ . One option is to choose a fixedrenormalisation scale, constant in time. This is formallycompletely valid, but may not provide a good approxi-mation of the effective potential for all times. Since theexact effective potential is independent of the choice of µ ,any large dependence on this parameter is a sign that theperturbative truncation is unreliable. The prescription istherefore that one should at each time choose µ so thatthe result depends as little as possible on its exact value.In particular, µ can be time-, field- and/or Hubble-ratedependent.We will do the next-best thing, and define µ = | M ( φ ) | e = | M ( φ, H, λ, m , ξ ) | e . (24)This is highly convenient, since the Coleman-Weinbergcontributions to both mean field and the entire quantumcorrection to the Friedman equations simply vanish. Theonly remaining effect of quantum corrections is that thecouplings in (5, 6, 7), in the expressions for the slow-rollformalism (3, 4, 8) and the observables (11, 12, 13) areidentical to the classical case, but with couplings thatrun with the RG-scale. The cost is that cancelling the Note that in the third equation for ξ , the 2-loop contribution isthe sum of terms proportional to λ . V ∝ ϕ V ∝ ϕ Class. 68 % Class. 95 % Quant. 68 % Quant. 95 % Planck 68 % Planck 95 % n s r . FIG. 1: The Planck allowed region in the n s - r plane(purple). Overlaid, the slow-roll result for monomial φ (green) and φ (orange) inflation. Also, non-slow-rollconfidence intervals for the classical φ + φ model(blue), and when quantum corrections are taken intoaccount (red).Coleman-Weinberg part, we have committed to a (time-dependent) choice of renormalisation scale, and we mayno longer vary it. We note that the fact that the samechoice of scale removes quantum corrections from bothfield equation and Friedmann equation is not generic. Wewill see an example below where it does not work out,when we consider a 2-field model of inflation.In order to solve the RG equations, we still need todefine the reference scale µ , where the parameters havevalues λ , m , ξ . We choose to pick the scale µ to cor-respond to the initial value of φ , φ , deep in the slow-rollinflationary regime. Then using the expression for µ de-fined above, we take µ = µ (0) = | M ( φ , H , λ , m , ξ ) | e . (25)This is an explicit expression for µ , if we for φ usethe slow-roll approximation (10) to determine H , ˙ φ and˙ H ∝ (cid:15) = 0. III. SINGLE-FIELD NUMERICAL ANALYSISA. Classical evolution, minimally coupled
In the classical limit, to get us started we can with rel-ative ease solve the slow-roll equations semi-analytically,in either of the monomial limits ( m = 0, ξ = 0) or( λ = 0, ξ = 0). Using the interval N ∗ = 50 to 60 wefor each set of parameter values (either m or λ ) com-pute n s and r , imposing the central value of the Planckconstraint on A s [1]log(10 A s ) = 3 . ± . . (26)This is shown as the two line segments in Fig. 1. Weconfirm the familiar result that classical φ inflation isfirmly outside the observationally allowed region (shownas purple contours), and φ -inflation marginally so.Overlaid are the results of a full numerical sampling,using classical dynamics where we allow to vary both λ , m (so not restricting to monomial inflation) and theinitial value φ in the intervals (see section VI for details) λ ∈ [10 − ; 10 − ] , m M ∈ [10 − ; 10 − ] , φ M pl ∈ [2; 30] . (27)Note that, although we quote parameters with subscript0, in the classical simulation, these do not RG-run andso are equal to λ , m (and shortly also ξ at all scales µ ). φ is the initial field value throughout. Also, we donot employ the slow-roll approximation for the field ormode evolution. The slow-roll parameters only enter aswe compute the observables to leading order in slow-roll(eqs. (11), (12), (13)). Rather than imposing the centralvalue of A s , we marginalize A s with a Gaussian distribu-tion of width as in (26). These regions are bounded byred lines in Fig. 1.We see that the allowed region in the n s − r is now anelongated ”banana”, nicely including the semi-analyticSR-intervals inside the 68% confidence regions. Thisgives us confidence in the numerical implementation, andalso suggests that the slow-roll approximation is in factrather good. With the given choice of the prior distribu-tions for the parameters, observables close to the classicalslow-roll results for monomial inflation are more likelythan others. In Fig. 2, we see that statistically, eitherthe λ or the m term dominates the potential and thesolution of the field equation behaves accordingly. B. Quantum evolution, minimally coupled
The red contours in Fig. 1 refer to a numerical scanof parameters ξ , m , λ , φ in the exact same rangesas before. But this time, the parameters run as theRG-evolution and scale µ changes in time as describedabove. We see that the classical and quantum regionsin the n s − r -plane are identical, up to statistical errors.We have checked that there is indeed convergence of thetwo regions as the statistics increases. In principle, onecould imagine having large quantum corrections shiftingindividual parameter points around in the n s − r plane,which just happens to create the very same overall dis-tribution. But we have checked that indeed for the indi- vidual points, the correction is tiny.To further investigate the quantum corrected be-haviour, we can analytically solve for the 1-loop runningof the couplings as a function of scale µ , finding λ = λ − λ π log (cid:104) µµ (cid:105) ,m = m (cid:16) − λ π log (cid:104) µµ (cid:105)(cid:17) / ,ξ −
16 = (cid:18) ξ − (cid:19) (cid:16) − λ π log (cid:104) µµ (cid:105)(cid:17) / . (28)An example is shown in Fig. 3, and we see that for thevery small couplings required by observations, the run-ning is very small indeed, unless the scale changes bymany orders of magnitude.To further exemplify the magnitude of the running,let us simply solve for the field evolution in the classicalslow-roll approximation, for the case ξ = 0, m = 0. Letus also, for the sake of argument assume that the initialvalue of the field is N e = 60 e-folds before the end ofinflation, the epoch of horizon crossing. Then we have achange of scale during the whole evolution (initial valueto end of inflation) of µ e µ = (cid:20) | M ( φ e , H e ) || M ( φ H , H H ) | (cid:21) = (cid:34) | λφ e M + λ (2 − (cid:15) e ) φ e || λφ H M + λ (2 − (cid:15) H ) φ H | (cid:35) , (29)where indices e and H denote the end of inflation and thehorizon crossing epoch, respectively, and where we haveused the relation R = H (12 − (cid:15) ) = λ φ (2 − (cid:15) ) . (30)Using (cid:15) = φ / (8 M ) for the classical λφ /
24 modelunder consideration, we find µ e µ = e − . (cid:39) , Monomial φ . (31)For λ = 10 − , using µ = µ /
45, the combina-tion 3 λ log[ µ/µ ] / (16 π ) featuring prominently in (28)is ± − , making the running completely negligible.This conclusion does not change upon using 2-loop RG-running, or considering a m φ / λ = 0, ξ = 0),where µ e µ = e − . (cid:39) , Monomial φ . (32)Note that choosing the origin of the running µ to bemuch deeper in the slow-roll regime (i.e. much before thehorizon crossing epoch) does not matter to the observ-ables n s , r , since they only depend on the value of the ( λ )/ log m / M pl2 n s ( λ )/ log m / M pl2 r FIG. 2: Scatter plot showing the dependence of n s and r on the ratio of the exponents of λ and m . The dominatingcontribution to the potential typically determines the behaviour of the model as ”almost” φ or φ monomialinflation. The horizontal bands indicate the ranges of these models in classical slow-roll for N ∗ between 50 and 60.Red and blue points represent data obtained with and without quantum corrections respectively. Included are onlydata points whose value for log(10 A s ) is within nine standard deviations of the Planck experimental value. - - - - - - - -
20 log ( μ / μ ) ( λ ϕ / λ ϕ , - ) - - - - - - - - - ( μ / μ ) ( m ϕ ² / m ϕ , ² - ) - - - - ( μ / μ ) ξ x FIG. 3: The relative 1-loop running couplings for the one-field model. Left: λ/λ −
1. Middle: m /m −
1. Right: ξ .For all three parameters, we have multiplied by 10 . For illustration, we have used λ = 10 − , m = 10 − M and ξ = 0.potential during and after the horizon crossing epoch.But it does change the mapping from observables to ba-sic variables λ , m , since they will have run for a whilebefore entering this epoch. Hence, only the running be-tween φ H and φ e matters. As we see, this is very small,and for the 1-field model, the running is the only quan-tum effect after choosing the pseudo-optimal RG-scale µ ( φ, H, ... ).As a result, in a single-field inflation model, quantumcorrections are very small. This follows from the runningof the couplings, which are all controlled by λ ; and thefact that (as we have seen) any other effect of quantumcorrections can be made to disappear by a convenientchoice of RG-scale µ . It is tempting to try to compensatethe smallness of the coupling by making µ very smallor large, so that the combination λ log( µ/µ ) is muchbigger. But that is precisely not allowed, since we havecommitted to a pseudo-optimal choice of µ in order totrust our perturbative approximation in the first place. We note that a further reduction of the µ -dependenceby finding the truly optimal expression for µ would onlyfurther emphasise this conclusion. We will return to thispoint shortly.We also note that although we are comparing the quan-tum evolution to a minimally coupled classical evolution,because ξ runs with scale, this is not really well-definedin the quantum case, except if setting the initial value tozero, ξ = 0, as we do here. C. Non-minimal coupling
Allowing now for (initial) non-minimal couplings in theinterval ξ ∈ [10 − ; 1] , (33) V ∝ ϕ V ∝ ϕ Class. 68 % Class. 95 % Quant. 68 % Quant. 95 % Planck 68 % Planck 95 % n s r . FIG. 4: The Planck allowed region in the n s – r plane(purple). Overlaid, the slow-roll result for monomial φ (green) and φ (orange) inflation. In addition,confidence intervals for the φ + φ + ξRφ model usingclassical dynamics (blue) and quantum correcteddynamics (red).we find Fig. 4. We perform the completely analogousnumerical procedure as for the minimally coupled case,running scans with and without quantum corrections.We again see a familiar result, that allowing for non-minimal coupling, the region consistent with the observed A s has a substantial overlap with the Planck-allowed re-gion. There is still a weaker correlation between n s and r along a diagonal in the plot. With our optimized choiceof the RG-scale, the quantum corrections do not signif-icantly modify the posterior distribution for the observ-ables. We found that allowing for non-zero initial ξ de-stroys the simple relationship of Fig. 2: Any relative sizeof λ and m can be compensated for by a choice of ξ to produce observables inside the Planck-allowed region.The probability distribution in fact takes its largest val-ues close to r = 0. D. Validating the choice of µ Because of the complicated dependence on dynamicalvariables ( φ , ˙ φ , H , ˙ H ), one cannot solve for the optimal µ in closed form µV eff dV eff dµ = 0 . (34)It is not even clear, that such a minimum exists. Also, itmakes little sense minimising in the complete variables + OOOOOOOO - - - - μ / M pl x 10 μ / V δ V / δ μ x FIG. 5: The pseudo-optimal scale (circles) at differenttimes (colours), and the relative dependence of thepotential on µ given sets of ( φ, ˙ φ, H, ˙ H ) along thetrajectory. Note the rescaling by 10 along both axes. µ -space, since the dynamical variables are not indepen-dent, but follow from a specific inflationary trajectory infield space. In Fig. 5 we show an example, where we pick4 times along a given trajectory in ( φ, ˙ φ, H, ˙ H )-space.For each of these, we compute (34) as a function of µ (dashed lines). We see that there is no obvious optimalvalue, but that the dependence on µ is very weak at alltimes during the trajectory. The time-dependent valueof the pseudo-optimal scale is indicated as circles. Thisscale moves towards smaller values during the evolution.This completes our analysis of the single-field model. IV. COUPLING TO A SPECTATOR FIELD
For any realistic model of inflation, the inflaton fieldmust be coupled to additional degrees of freedom, andultimately to the fields of the Standard Model. Becausethese have quantum fluctuations, their presence is en-coded in the effective action of the inflaton itself. Butnow, because classically the coupling between the twois unconstrained by CMB observables, we can in prin-ciple imagine choosing it arbitrarily large, O (1). Thereare some constraints due to the reheating mechanism,and how it affects the last few e-folds at the end of in-flation. We will not take this complication into accounthere, since it requires to establish a full field theory sim-ulation. But even for couplings much below unity, ourexpectation is that the allowed values of the parameters(say, λ φ , m φ , ...) are very much dependent on the in-clusion of quantum corrections and the interactions withthese other degrees of freedom.A simple example of this is to consider two scalar fields,the inflaton φ and a representative of all the other degreesof freedom σ . Note that σ is not intended to be a curva-ton. The CMB and the expansion of the Universe bothoriginate from the dynamics of φ . The action reads S = (cid:90) d n x √− g (35) × (cid:20) M Rf ( φ, σ ) + 12 ∂ µ φ ∂ µ φ − m φ φ − λ φ φ − g φ σ + 12 ∂ µ σ ∂ µ σ − m σ σ − λ σ σ (cid:21) , with f ( φ, σ ) = 1 − ξ φ φ M − ξ σ σ M , (36)and where for consistency, we have again allowed fornon-minimal coupling to gravity. We will stipulate that φ provides the dominant energy component through itsmean field being displaced from the potential minimum,whereas σ will be taken to be in its vacuum around σ = 0.In this case, the classical A. Quantum corrections, two fields
Computing the 1-loop effective potential, with 1-loopRG running, we have V eff = 12 m φ φ + 12 ξ φ Rφ + 124 λ φ φ (37)+ 164 π M φ ( φ ) (cid:34) log | M φ ( φ ) | µ − (cid:35) + 164 π M σ ( φ ) (cid:20) log | M σ ( φ ) | µ − (cid:21) , (38)where as advertised we have already imposed that σ = 0.But, contrary to the classical limit, at the quantum levelthe presence of the spectator field σ is apparent in thelogarithmic corrections to the effective potential. We nowhave two effective mode masses M φ ( φ ) = m φ + λ φ φ + ( ξ φ − / R, (39) M σ ( φ ) = m σ + g φ + ( ξ σ − / R. (40)We expect that λ φ is small to allow for slow-roll infla-tion. On the other hand, g need not be, and so gener-ically M σ (cid:39) gφ /
2, much larger than the Hubble rate H . Hence for the purposes of CMB observables, σ is aheavy field and does not contribute to the density pertur-bations. This is as expected and a useful simplification. σ = 0 is a solution also of the quantum evolution, had we writtendown and solved for the σ equation of motion. The equation of motion for φ follows by variation of theeffective action,¨ φ + 3 H ˙ φ + (cid:0) m φ + ξ φ R (cid:1) φ + 16 λ φ φ (41)+ λ φ φM φ ( φ )32 π (cid:32) log | M φ ( φ ) | µ − (cid:33) + gφM σ ( φ )32 π (cid:18) log | M σ ( φ ) | µ − (cid:19) = 0 . We can now choose a pseudo-optimal RG scale µ in sucha way that the two logarithmic terms cancel out. Wehave µ M = e − (42) × exp λ φ M φ ( φ ) log (cid:104) | M φ ( φ ) | M (cid:105) + gM σ log (cid:104) | M σ ( φ ) | M (cid:105) λ φ M φ ( φ ) + gM σ ( φ ) . Again, this is not the truly optimal choice, but is a veryconvenient one. This also trivially means that (cid:18) log | M σ ( φ ) | µ − (cid:19) = − λ φ M φ ( φ ) gM σ ( φ ) (cid:32) log | M φ ( φ ) | µ − (cid:33) . (43)As for the single-field case, the Friedmann equations donot follow from variation of the effective action, but froma separate computation, varying the classical action withrespect to g µν . There are now two fields, with each theircontribution (even after setting σ = 0). We have T Qµν = − g µν H π (cid:20) A φ (cid:32) log | M φ ( φ ) | µ − (cid:33) + A σ (cid:18) log | M σ ( φ ) | µ − (cid:19) (cid:21) , (44)with A φ = (cid:2) δ φ − δ φ (cid:15) − δ φ − (cid:15) +12 ξ φ (2 − δ φ + (cid:15) − δ φ (cid:15) ) (cid:3) , (45) A σ = (cid:2) δ σ − δ σ (cid:15) − δ σ − (cid:15) +12 ξ σ (2 − δ σ + (cid:15) − δ σ (cid:15) ) (cid:3) . (46)The (cid:15) is defined as before, and we again have the slow-roll-like quantities δ φ = m φ + λ φ φ + ξ φ RH , δ σ = m σ + g φ + ξ σ RH . (47)Note however that δ σ is not small in the slow-roll sense. σ is not a light field. It is clear that the choice of µ ,eq. (42), no longer makes the quantum corrections vanishin the expression for the energy density (as they did forthe 1-field case). However, it does allow us to replace onelogarithm by another, in accordance with (43), to get T Qµν = − g µν H π (cid:32) log | M φ ( φ ) | µ − (cid:33) × (cid:18) A φ − A σ λ φ M φ ( φ ) gM σ ( φ ) (cid:19) , (48)It is worth looking at the size of this object, in the sim-plified case of ξ φ = ξ σ = m σ = λ σ = 0 (also neglecting,for now, their reappearance due to running). Then, ne-glecting all SR-sized quantities, we have T Qµν (cid:39) − g µν H π (cid:32) log | M φ ( φ ) | µ − (cid:33) (49) × (cid:26) − g φ H + 2 gφ H (cid:27) λ φ ( m φ + λ φ φ ) gφ , (cid:39) g µν λ φ g π (cid:32) m φ φ + λ φ φ (cid:33) (cid:32) log | M φ ( φ ) | µ − (cid:33) . This may be compared to the tree-level contributions (cid:39) g µν (cid:18) m φ φ + λ φ φ (cid:19) , (50)which is seen to dominate as long as gλ φ π (cid:32) log | M φ ( φ ) | µ − (cid:33) (cid:28)
12 or 112 . (51)This can easily be accommodated. We emphasise thatthis criterion follows from the choice of RG-scale µ , sincewithout the substitution of (43), the prefactor of the loga-rithm would have been ( gφ ) , which would dominate thetree-level contribution as soon as g (cid:39) λ φ . And thisin turn would be possible for quite sensible values of g .So, although the RG scale does not allow us to cancel outthe quantum corrections completely in this case, it doessuppress them to a subleading contribution, compared tocontributions at tree level. We will confirm this a poste-riori below, and it means that we can again identify theleading quantum corrections to be the RG-running of thecouplings. B. RG-running for two fields
There are now seven RG-running couplings m φ , m σ , λ φ , λ σ , ξ φ , ξ σ , g , and we derive a set of coupled 1-loop RG equations,1 µ dλ φ dµ = 3(4 π ) (cid:2) λ φ + g (cid:3) , µ dλ σ dµ = 3(4 π ) (cid:2) λ σ + g (cid:3) , µ dm φ dµ = 1(4 π ) (cid:2) λ φ m φ + gm σ (cid:3) , µ dm σ dµ = 1(4 π ) (cid:2) λ σ m σ + gm φ (cid:3) , µ dξ φ dµ = 1(4 π ) [ λ φ ( ξ φ − /
6) + g ( ξ σ − / , µ dξ σ dµ = 1(4 π ) [ λ σ ( ξ σ − /
6) + g ( ξ φ − / , µ dgdµ = 1(4 π ) g [ λ φ + λ σ ] . (52)We are mostly interested in the dependence on the cou-pling g , but we see that λ φ , λ σ and g source each othersRG-running, and so if g is non-zero, they all are. Sim-ilarly, only if both masses m φ and m σ vanish, do theyremain zero. Finally, if both ξ φ and ξ σ are equal to theirconformal value 1 / λ φ starts outwith a value of (cid:39) ( − , − , − , in accordance with theclassical observational constraints, it grows semi-linearlyas 3 / (4 π ) g × log[ µ/µ ]. Hence even for a reasonable,perturbative value of g = 10 − , λ φ will have grown to > − with a scale-change of log[ µ/µ ] = O (1). Thiswould naively suggest that the system is driven out ofslow-roll. It is therefore highly conceivable that the in-clusion of quantum fluctuations puts strong constraintson the viable values of g . And at the same time, a care-ful choice of g may allow otherwise ruled-out scenarios(such as quartic inflation) to fit observations. We willinvestigate this further below. In Fig. 6 we show one ex-ample of the running couplings, using m φ, = 10 − M , λ φ, = 10 − and g = 10 − .We observe that all parameters become non-zero. Theleft panel shows the relative change of the inflaton self-coupling. We see that it can grow by a factor 1 − µ/µ ] = −
4. Inthis example, we have used the maximum value of thisinitial coupling and the largest value of g , suggestingthat quantum corrections may become important, butneed not be. We will see shortly what the effect is.We note in passing that the non-minimal coupling re-mains very small, and that the relative change in theinflaton mass parameter m φ is very small. Hence, theprimary effect of the RG running is for the inflaton self-coupling to change significantly during the time evolu-tion. The inflaton self-coupling λ σ grows as large as λ φ ,but does not enter in the expressions for the observables.0 - - - - - - - -
20 log ( μ / μ ) λ ϕ / λ ϕ , - - - - - ( μ / μ ) ( m ϕ ² / m ϕ , ² - ) - - - - ( μ / μ ) ( g / g - ) FIG. 6: The 1-loop running couplings for the two-field model. Whereas m φ and g are again multiplied by 10 , therelative running of λ φ is not. For illustration, we have used λ φ, = 10 − , m φ, = 10 − M and g = 10 − . Theremaining parameters λ σ , m σ , ξ φ,σ also run (not shown). V. TWO-FIELD NUMERICAL ANALYSISA. Individual trajectories
To illustrate the impact of quantum effects on the cos-mological evolution we first choose 3 examples for whichinflation is successful. The initial parameters at µ = µ for these cases areBM 1: m φ, = 6 . · − M , λ φ, = 5 . · − ,g = 6 . · − , φ = 25 M pl , BM 2: m φ, = 1 . · − M , λ φ, = 2 . · − ,g = 2 . · − , φ = 21 M pl , BM 3: m φ, = 1 . · − M , λ φ, = 4 . · − ,g = 2 . · − , φ = 30 M pl , (53)and ξ φ, = λ σ, = ξ σ, = 0 for BM 1 - BM 3. We choose N ∗ = 55 as the pivot scale everywhere and (cid:15) = 1 as thecondition for the end of inflation.We implement the numerics as for the one-field model.The dynamical variables are the same, but the evolutionequations and solving for the time-dependent µ is moreinvolved. The classical limit is identical to the one-fieldcase (since we are postulating that the second field doesnot contribute to the energy density of the Universe, norto the density perturbations).For the 3 parameter sets we obtain the following nu- merical values for the observables:BM 1: A s = 3 . · − , δA s = − . .n s = 0 . , δn s = 0 . .r = 0 . , δr = − . , BM 2: A s = 7 . · − , δA s = − . · − .n s = 0 . , δn s = 5 . · − .r = 0 . , δr = 1 . · − . BM 3: A s = 9 . · − , δA s = − . .n s = 0 . , δn s = 0 . .r = 0 . , δr = 0 . .δA s , δn s and δr represent the relative changes comparedto the classical results. We see that the largest differencesin these examples are at the percent level.As described, we update the renormalisation scale toremove the quantum contributions to the field-equation.We find a posteriori that the remnant corrections to theFriedmann equations are negligible with this choice of µ (see Fig. 7). We show the magnitude along BM1-3 ofthe logarithmic corrections to the Friedmann equations,relative to the non-logarithmic contributions. When theRG-scale is not adjusted over time as (42) (dashed lines)the contributions are small, but may still be a few percentor more. By adjusting the RG-scale (full lines), we canreduce this to one part in 10 − or less. And so althoughwe are not able to identically cancel out the logarithmsas for the 1-field model, choosing the RG-scale wisely is avast improvement on even the ”best” constant RG-scale.We therefore ignore these contributions keeping only theRG-running of the couplings according to equations (52).The relative change of m φ , λ φ , and g due to RG-running is shown in Fig. 8. For larger values of g thischange can be noticeable. They are reflected in the quan-tum evolution of the field shown in Fig. 9.1 - - - N T Q / T C BM 1 ( opt ) BM 1BM 2 ( opt ) BM 2BM 3 ( opt ) BM 3
FIG. 7: Relative size of the quantum corrections to the Friedman equations as a function of the evolved number ofe-folds. The dashed lines correspond to these contributions with constant RG-scale, while the full lines show theirsize if the RG-scale is updated (in both cases computed along the trajectory obtained with only RG-runningcorrections). - - - - N | m ϕ / m ϕ , - | - - N | λ / λ - | - - - - N | g / g - | BM 1BM 2BM 3
FIG. 8: Relative change of some of the two-field model parameters as a function of the evolved number of e-folds N for different initial parameters (BM 1 - BM 3). Further parameters that have zero initial values are not shown butrun as well. B. Full parameter scan
It is tempting to search for parameter sets that maxi-mize the effect of the quantum contributions. However,we want to convey an unbiased picture of their relevance.As before, we therefore sample the model parametersand initial conditions and evolve the non-slow roll equa-tions of motion, computing the basic CMB observables.Marginalizing over A s , we again generate a region in the n s − r plane. To reduce the sampling parameter spacesomewhat, we take λ σ, = 0, m σ, = 0, ξ φ, = 0 and ξ σ, = 0 throughout. Because they all run, they onlyvanish at φ . This leaves as initial input λ φ, , m φ, andthe coupling g . We again take the ranges (27), and add g ∈ [10 − ; 10 − ] . (54)Fig. 10 shows the two-field classical and quantum re-gions consistent with observations in the n s − r -plane.As before, the slow-roll monomial results are indicatedby the orange and green line segments, and the Planckresults are marked with purple lines. For the presentsweep, ξ φ, is zero. As a consequence, the classical re-2 N ϕ N H N μ BM 1BM 1 ( class ) BM 2BM 2 ( class ) BM 3BM 3 ( class ) FIG. 9: Evolution of field value, Hubble rate and the inferred renormalisation scale. For large values of g (BM 1) theRG-running of the parameters can have a noticeable effect on the cosmological evolution. V ∝ ϕ V ∝ ϕ Class. 68 % Class. 95 % Quant. 68 % Quant. 95 % Planck 68 % Planck 95 % n s r . FIG. 10: The confidence regions for classical andquantum evolution the two-field case in the n s − r plane.Overlaid the Planck allowed range and the slow-rollmonomial benchmarks (orange and green lines).sult is the ”banana”-shaped φ + φ region from Fig. 1.Without the RG-running this agreement with the classi-cal limit of the single-field model should be exact and, if g is small, it should hold approximately, since the quantumcontributions by the other non-vanishing parameters are tiny.We see that the overall differences between the evo-lution with both small g and large g are small. Thisis somewhat surprising, since we have seen that at leastfor trajectories of the BM1-type, both evolution and ob-servables are shifted by some percent. Apparently, whensampling over the whole range, one recovers the sameallowed region, even though individual trajectories (cor-responding to a given parameter set) are moved aroundwithin this region.One may speculate that even larger values of g will leadto larger effects such that the n s − r contour is broughtinto overlap with the Planck allowed region. However,we find that for g (cid:38) − the observed value for A s isnot met any more (see Fig. 11). And as (cid:38) − , infla-tion is ruined altogether by the running of λ φ sourced bylarge g . It is possible that carefully tuning to the narrowregion around g (cid:39) − one may find some singular caseswith large corrections to the observables, but by a flat-logarithmic sampling as we do here, such a region doesnot show up. C. Validating the choice of µ Along a trajectory in ( φ , ˙ φ , H , ˙ H )-space, we againcompute the relative dependence on the scale (34). InFig. 12 is then the analogue of Fig. 5, but for two cou-pled fields. We see that the dependence on µ is muchlarger than for the 1-field model (Fig. 5 was scaled upwith 10 , Fig. 12 is not), but still at the percent level formost of the range shown. Our time-dependent, pseudo-optimal choice of scale (circles) nicely traverse the range3 - - - - - - - -
505 log ( g ) l og A s - - - - - - - ( g ) n s - - - - - - - ( g ) r FIG. 11: Scatter plots showing the observables as a function of g . For large values we see a deviation from theregion given by the slow-roll results for monomial inflation. However, for g (cid:38) − the data fails to satisfy theexperimental constraints on A s indicated by the solid purple horizontal line. For (cid:38) − inflationary evolution doesnot occur at all. OOOOOOOO - - - - - μ / M pl μ / V δ V / δ μ FIG. 12: The pseudo-optimal scale at different times(circles), and the potential as a function of µ given setsof ( φ, H, ˙ H ) along the trajectory.of scales, never coming near the singular region near thex-axis. In fact we see that as the pseudo-optimal scale de-creases, the correspondingly colour-coded µ dependenceof V eff (red to orange) becomes more shallow. We havenot observed a trajectory where the pseudo-optimal scalehas caught up with the large µ -dependence region. Weconclude that our prescription for the choice of scale,apart from being very convenient also ensures small µ -dependence. VI. NUMERICAL IMPLEMENTATION
In this section we provide some more details on howwe solve the field- and Friedmann equations and how wesample the parameters. Our numerical implementationis based on the public Fortran package MultiModeCode[22]. It is suited to the computation of observables formulti-field inflation models in the δN -formalism, or bysolving the mode equations directly. In the present workwe make use of the former feature only. We however have to adapt the code significantly, inorder to support the non-minimal coupling to gravity aswell as the quantum corrected field (16) and Friedmannequations (17) (or equations (41) and (44) for the two-field model) that take a non-standard form. They repre-sent a coupled system of ordinary differential and alge-braic (constraint) equations. This system determines theevolution of mean field values and the Hubble rate in timeor, after a corresponding transformation, in the numberof e-folds since the beginning of inflation. Usually, the al-gebraic first Friedman equation is used to explicitly solvefor H .With quantum corrections, however, its right-handside depends in a complicated way on H . This is trueeven if the logarithms in (16) and (17) are eliminatedby choice of the renormalisation scale, because µ thenbecomes a function of H (as well as φ and ˙ φ ). Recallthat, in each step, µ is determined as the solution of thealgebraic equation (25) for µ .To avoid the numerically solving the Friedmann equa-tion, which would also need to be repeated in each timestep of the evolution, we choose to solve the second Fried-man equation for dH/dN . In the single-field case, thisis possible explicitly if the quantum terms are made zerothrough the choice of µ described above. In the two-fieldcase, the choice of µ that eliminates the direct quan-tum contributions to the mean field equation leaves anon-vanishing quantum contribution to both Friedmannequations. As argued above, for the purposes of thepresent paper we can neglect these terms in numericalcomputations.For both models, we then arrive at the first order sys-tem (with m φ = m , λ φ = λ and ξ φ = ξ in the 1-field4case) φ (cid:48) = v , (55) v (cid:48) = H (cid:48) vH − H (cid:18) H v + (cid:0) m φ + ξ φ H ( H (cid:48) + 2 H ) (cid:1) φ + 16 λ φ φ (cid:19) ,H (cid:48) = H −
48 ( ξ φ (6 ξ φ − φ + 1) × (cid:2) H (6 ξ φ (1 − ξ φ ) φ − ξ φ φv + (4 ξ φ − φ (cid:48) − φ (cid:0) λ φ (1 − ξ φ ) φ + m φ (12 − ξ φ ) (cid:1) (cid:3) , where the primes denote the derivative with respect tothe number of e-folds, φ (cid:48) = dφ/dN etc. The cost of thisapproach is that we need to solve the differential equationfor H alongside hose for the fields. All model parametersin (55) are functions of µ = µ ( H, φ, φ (cid:48) ) which needs tobe computed numerically in each step. This solution of(24) or (42) determines the overall numerical costs.The first Friedman equation is used, together with thefield-equation, in slow-roll approximation to determinethe initial conditions self-consistently:0 = 3 H φ (cid:48) (56)+ (cid:0) m φ, + ξ φ, H ( H (cid:48) + 2 H ) (cid:1) φ + 16 λ φ, φ , H = H φ (cid:48) + m φ, φ + λ φ, φ + 6 H φ (cid:48) φ ξ φ, − φ ξ φ, ,H (cid:48) = H −
48 ( ξ φ, (6 ξ φ, − φ + 1) × (cid:2) H (6 ξ φ, (1 − ξ φ, ) φ − ξ φ, φ φ (cid:48) + (4 ξ φ, − φ (cid:48) − φ (cid:0) λ φ, (1 − ξ φ, ) φ + m φ, (12 − ξ φ, ) (cid:1) (cid:3) . Given the initial model parameters and an initial valuefor the field, φ , this non-linear algebraic system deter-mines H , φ (cid:48) and H (cid:48) . It can be solved explicitly, but wesolve it numerically for accuracy reasons.The determination of consistent slow-roll initial condi-tions as well as that of the renormalisation scale, requiressolving non-linear systems of equations. We have linkedthe MultiModeCode package with Mathematica’s WSTP(former MathLink library). This allows to reach the levelof precision required to not pollute MultiModeCode’sadaptive solving of the system of differential equationsthat govern inflation and to expose the typically smallquantum effects. We can perform these computations at Combined with the fact that the field equation is no longer deter-mined by the classical potential that also governs the potentialslow-roll parameters this constitutes the main change that needsto be made to an inflation solver like MultiModeCode. arbitrary precision and need not re-express the lengthyquantum contributions in numerically stable form for themany possible hierarchies of the parameters.Throughout the paper we have chosen flat prior dis-tributions for the logarithms of all model parameters at µ = µ , each on the intervals quoted in the sections. Forthe field φ we generate initial values on the interval [2; 30]obeying a flat non-logarithmic distribution. As the pivotscale k ∗ we take the value k ∗ = 0 .
002 everywhere. N ∗ varies on the flat prior [50; 60] throughout. To determinethe end of inflation we use the condition (cid:15) = 1. Sampledparameter sets for which not sufficiently many e-folds areachieved or for which inflation continues indefinitely areignored.Based on the obtained numerical values for the ob-servables A s , n s and r we estimate their multi-variatedistribution P ( A s , n s , r ) and marginalize it with a Gaus-sian for log(10 A s ), centred at the Planck experimen-tal value, to obtain P ( n s , r ). The confidence regions inFig. 1, 4 and 10 are given by the contour lines for thevalues P α ( α = 95 . , . (cid:90) + ∞−∞ θ ( P ( n s , r ) − P α ) P ( n s , r ) dn s dr = α . (57)As explained, the parameters are constrained by therequirement for successful inflation. To perform aBayesian parameter estimation a Monte-Carlo samplingmethod would need to be employed, given the high-dimensionality of the parameter space. In view of themoderate influence of the quantum contributions in theconsidered models, and the entailed additional numericalcomplexities, we have not attempted this in the presentwork, relying on information from scatter plots. Some ofthese are displayed above. VII. CONCLUSION
We have studied quantum corrections to inflation, andtheir impact in ruling certain models in and out of thePlanck-allowed region in n s − r space. Specifically, weconsidered a single-field model with quadratic and quar-tic terms, coupled non-minimally to gravity as well asa two-field model in which the second field has a por-tal coupling with the inflaton field. We simplified thecomputations by taking the second field to be in its vac-uum, mimicking spectator fields coupled to the StandardModel. The quantum corrections enter in the form of theRG-running of the model parameters and quantum cor-rections to the energy momentum tensor that contributeto field and Friedmann equations.The size of the latter depends on the choice of therenormalisation scale. We show that they can be madezero in the single-field and very small in the two-fieldcase, leaving effectively only the RG-running to be con-sidered. Although not the truly optimal choice of µ , we argued that the precise value was not crucial.5For the single-field model, the smallness of the infla-ton self-coupling made the running negligible altogether.Whereas in the two-field model, the running was stillsizeable.At the classical level, we confirmed that non-minimalcouplings to gravity can bring the considered models intoagreement with Planck measurements. In order to ob-tain an unbiased statement about the relevance of thequantum corrections caused by the RG-running of theparameters, we sampled the model parameters on broadintervals compatible with existing constraints. We foundthat the quantum corrections are negligible in the single-field case.In the second case with two fields, we found that theRG-running can influence the trajectories of the field no-tably if the portal coupling takes large enough values.However, the overall effect on the observables does hardlyaffect the contours of the posterior probability distribu-tion for the observables. It seems that individual trajec-tories are swapped around inside the contours, withoutsubstantially altering them. Interestingly, we found thatlarge values of the portal coupling tend to ruin inflationaltogether putting a limit of g < − , up to details ofthe scale µ ( φ ) where g is introduced. This bound also restricts the size of the quantum corrections.In conclusion, a correct and reliable implementationand analysis of quantum corrections to inflation involvesconsidering RG-running, allowing for a time- and field-dependent RG-scale, including curvature corrections andimposing observational limits on A s . We find that φ + φ -inflation coupled to a spectator field is as disfavouredat the quantum level as for the classical approximation.We did not do explicit scans for ξ (cid:54) = 0 for the two-fieldmodel, but suspect that a similar conclusion may apply.The obvious next steps include adapting our procedureto other models of inflation, as well as including IR effectsarising from resummations of diagrams (see for instance[19]). Of particular interest would be to consider relax-ing the semi-classical approach to include both scalar andtensor metric degrees of freedom in our quantum treat-ment [21]. This poses a number of other challenges to dowith the running of gravitational couplings and renor-malizability. ACKNOWLEDGMENTS
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