Is asymptotically safe inflation eternal?
Jan Chojnacki, Julia Krajecka, Jan H. Kwapisz, Oskar S?owik, Artur Str?g
PPrepared for submission to JCAP
Is asymptotically safe inflationeternal?
J. Chojnacki, , ∗ J. Krajecka, J. H. Kwapisz, O. Slowik, , , A.Strag Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668Warsaw, Poland Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2,02-097 Warsaw, PolandE-mail: [email protected], [email protected],[email protected], [email protected], [email protected]
Abstract.
Recently, based on swampland considerations in string theory, the (no) eternalinflation principle has been put forward. The natural question arises whether similar con-ditions hold in other approaches to quantum gravity. In this article, the asymptotic safetyhypothesis is considered in the context of eternal inflation.As exemplary inflationary models the SU(N) Yang-Mills in the Veneziano limit and variousRG-improvements of the gravitational action are studied. The existence of UV fixed pointgenerically flattens the potential and our findings suggest no tension between eternal infla-tion and asymptotic safety, both in the matter and gravitational sector in contradistinction tostring theory. Moreover, the eternal inflation cannot take place in the range of applicabilityof effective field quantum gravity theory.We employ the analytical relations for eternal inflation to some of the models with singleminima, such as Starobinsky inflation, alpha-attractors, or the RG-improved models and ver-ify them with the massive numerical simulations. The validity of these constraints is alsodiscussed for a multi-minima model. * Corresponding author. a r X i v : . [ g r- q c ] J a n ontents Our Universe consists of 4 fundamental forces. Three of these forces have been consistentlydescribed on the quantum level and combined into the Standard Model of particle physics.Only quantum gravity seems to be elusive and has not been fully described in terms of quan-tum theory. This is not only because gravity is power counting non-renormalizable but alsodue to the fact that direct quantum gravity regime cannot be accessed experimentally (forexample an accelerator measuring the quantum gravity effects would have to be big as ourSolar System).In recent years an alternative strategy has been put forward, namely one formulates a funda-mental quantum gravity theory and then tests, which of the low energy effective theories canbe UV completed by this quantum gravity model. In string theory this goes under the nameof swampland conjectures [2, 3]. Recently widely discussed is so-called de-Sitter conjecture[4, 5] which states that string theory cannot have de-Sitter vacua and is in tension with singlefield inflation [6, 7]. There seems to be also a tension between standard S-matrix formulationof quantum gravity and existence of stable de-Sitter space [8–11]. However it is not estab-lished whether asymptotic safety admits a standard S-matrix formulation [12] due to fractalspacetime structure in the deep quantum regime [13, 14] In line of these swampland criteriathe no eternal inflation principle has been put forward [1], see also the further discussions onthe subject of eternal inflation [15–26].On the other hand, the theory of inflation is a well-established model providing an answer toproblems in classical cosmology, such as the flatness problem, large-scale structures formation,homogeneity, and isotropy of the universe. A handful of models is in an agreement with theCMB observations. In the inflationary models, quantum fluctuations play a crucial role inprimordial cosmology, providing a seed for the large-scale structure formation after inflation– 1 –nd giving a possibility for the eternally inflating multiverse. Initial fluctuations in the earlyuniverse may cause an exponential expansion in points scattered throughout the space. Suchregions, rapidly grow and dominate the volume of the universe, creating ever-inflating, dis-connected pockets. Since so far there is no way to verify the existence of the other pockets,we treat them as potential autonomous universes, being part of the multiverse.In the light of this tension between string theory and inflationary paradigm [1], one can beinterested in how robust are the swampland criteria for the various quantum gravity models.In accordance with the inflation theory, we anticipate that the dynamics of the universe arebeing determined by the quantum corrections to the general relativity stemming from theconcrete UV model. The effective treatment led Starobinsky to create a simple inflationarymodel taking into account the anomaly contributions to the energy-momentum tensor.As pointed out by Donoghue [27] below the Planck scale, for quantum gravity one can safelytake the effective field theory perspective. Yet these quantum gravity effects can be importantbelow the Planck scale by the inclusion of higher dimensional operators. The gravitationalconstant G N has a vanishing anomalous dimension below the Planck scale and various log-arithmic corrections to the R have been considered, capturing the main quantum effects[28–31]. Yet in order to get the correct 60 e-fold duration of inflationary period one has topush the scalar field value in the Einstein frame beyond the Planck mass [1]. Furthermoremost of these models do not possess a flat potential limit (either diverge or have a runawaysolutions), suggesting that eternal inflation can be investigated only if one takes into accountthe full quantum corrections to the Starobinsky inflation.In the effective field theory scheme, the predictive power of the theory is limited, as the de-scription of gravity at transplanckian scales requires fixing infinitely many coupling constantsfrom experiments. The idea of asymptotic safety [32] was introduced by Stephen Weinbergin 1978 as a UV completion of the quantum theory of gravity. The behavior of an asymptot-ically safe theory is characterized by scale invariance in the high-momentum regime. Scaleinvariance requires the existence of a non-trivial Renormalization Group fixed point for di-mensionless couplings. There are many possible realizations of such non-trivial fixed pointscenario, such as the canonical vs anomalous scaling (gravitational fixed point [33–36]) andone-loop vs two-loop contributions or gauge vs Yukawa contributions, see [37] for further de-tails and [38] for current status of asymptotically safe gravity.The existence of an interacting fixed point and hence the flatness of the potential in the Ein-stein frame led Weinberg to discuss [39] cosmological inflation as a consequence of Asymptot-ically Safe Gravity, see also [40, 41] for discussion of AS cosmology. Following this suggestion,we study two types of models.The first type relies on the RG-improvement of the gravitational actions and is based on theasymptotic safety hypothesis that gravity admits a non-trivial UV fixed point. Since asymp-totically safe gravity flattens the scalar field potentials [42], one can expect that it will resultin the eternal inflation for large enough initial field values. On the other hand, RG-improvedactions can serve as a UV completion of the Starobinsky model. One should also note thatasymptotically safe swampland has been vastly studied [43–69].The other model relies on the non-trivial fixed point in the pure matter sector governed bythe Yang-Mills dynamics in the Veneziano limit [70, 71], see also [72–75, 75]. In this model,we have uncovered a new type of eternal inflation scenario relying on tunneling to a falsevacuum - in the opposite direction as it was considered in the old-inflation proposal [76].In contradistinction to string theory, the couplings in the asymptotic safety paradigm arepredicted from the RG-flow of the theory and their fixed point values rather than as vacuum– 2 –xpectation values (vev’s) of certain scalar fields. Hence, the asymptotically safe eternally in-flating multiverses landscape is much less vast than the one stemming from the string theory,making these models much less schismatic [77]. Finally, let us note that asymptotic safety canargue for the homogeneous and isotropic initial conditions on its own using the finite actionprinciple [78].Our work is organized as follows. In Chapter 2 we introduce the idea of eternal inflationand multiverse. We discuss necessary conditions for eternal inflation to occur based on theFokker-Planck equation. In Chapter 3 we show, how the developed tools work in practice withthe two popular inflationary models. Chapter 4 is devoted to the presence of eternal inflationin Asymptotically Safe models. In Chapter 5 the results are discussed and concluded. In this section we discuss, under what circumstances the inflation becomes eternal. Ourdiscussion follows closely [1].
Consider a scalar field in the FLRW metric S = (cid:90) d √− g (cid:18) M P l R + 12 g µν ∂ µ φ∂ ν φ − V ( φ ) (cid:19) , (2.1)with φ ( t, (cid:126)x ) = φ ( t ) one obtains the following equations of motion: ¨ φ + 3 Hφ + ∂V∂φ = 0 , H M P = 13 (cid:18)
12 ˙ φ + V ( φ ) (cid:19) , (2.2)which in the slow-roll approximation become [79]: H ˙ φ + ∂V∂φ ≈ , H M P l ≈ V ( φ ) . (2.3)Inflation ends once one of the so-called slow-roll parameters becomes of order one (cid:15) (cid:39) M P (cid:18) V ,φ V (cid:19) , η (cid:39) M P V ,φφ V , (2.4)and enters the oscillatory, reheating phase. The standard treatment of eternal inflation relieson the stochastic inflation approach [80]. One splits the field into classical background andshort-wavelength quantum field φ ( t, (cid:126)x ) = φ cl ( t, (cid:126)x ) + δφ ( t, (cid:126)x ) . (2.5)Due to the fact that action is quadratic in the fluctuations, their spatial average over the Hub-ble volume is normally distributed. Hence from now on we shall assume that both backgroundand fluctuations are homogeneous, which is the standard treatment of eternal inflation (if nototherwise specified). In the large e-fold limit, equation of motion for the full field takes formof slow-roll equation with additional classical noise term [1, 81, 82], known as the LangevinEquation: H ˙ φ + ∂V∂φ = N ( t ) , (2.6)– 3 –here N ( t ) is a Gaussian distribution with mean equal 0 and variance σ = H t π [83]. Thenthe probability density of the inflaton field is then given by the Fokker-Planck equation [1]: ˙ P [ φ, t ] = 12 (cid:18) H π (cid:19) ∂ P [ φ, t ] ∂φ∂φ + 13 H ∂ i (cid:0) ∂ i V ( φ ) P [ φ, t ] (cid:1) , (2.7)where ˙ P [ φ, t ] := ∂∂t P [ φ, t ] . To understand better the Fokker-Planck equation, let us now briefly discuss the analyticalsolutions.
Case 1. Constant potential V ( φ ) = V , (2.8)the Fokker-Planck equation has a solution given by a Gaussian distribution: P [ φ, t ] = 1 σ ( t ) √ π exp (cid:34) − ( φ − µ ( t )) σ ( t ) (cid:35) , (2.9)with µ ( t ) = 0 , σ ( t ) = H π t. (2.10)A delta-function distribution initially centered at φ = 0 will remain centered at φ = 0 for alltime. It will however, spread out by the amount σ (cid:0) t = H − (cid:1) = H/ π after a Hubble time.This represents the standard “Hubble-sized” quantum fluctuations that are well-known in thecontext of inflation, famously imprinted in the CMB and ultimately seeding the observedlarge-scale structure. Case 2. Linear potential
For the linear hilltop model the potential is given by V ( φ ) = V − αφ. (2.11)Fokker-Planck equation is again solved by the Gaussian distribution (2.9) with: µ ( t ) = α H t, σ ( t ) = H π t. (2.12)The time-dependence of µ ( t ) is due to the classical rolling of the field in the linear potential.The time-dependence of σ ( t ) is purely due to Hubble-sized quantum fluctuations, and it pre-cisely matches the result in the constant case. In general, for a linear and quadratic potentialthe equation simplifies to the heat equation, hence the solutions are gaussian. Furthermore,the majority of the inflationary potentials can be approximated by a constant at the largefield values and expanded around the infinity.– 4 – .3 Eternal inflation conditions Given an arbitrary field value φ c , one can ask what is the probability that quantum field φ = φ ( t ) is above this value: Pr[ φ > φ c , t ] = (cid:90) ∞ φ c dφP [ φ, t ] . (2.13)Since the distribution is Gaussian, then for φ c large enough the P r [ φ > φ c , t ] can be approx-imated by an exponential decay: Pr[ φ > φ c , t ] ≈ C ( t ) exp( − Γ t ) , (2.14)where C ( t ) is polynomial in t and all of the dependence on φ c is contained in C ( t ) . Then itseems that inflation cannot last forever since lim t →∞ Pr[ φ > φ c , t ] = 0 . (2.15)However, there is an additional effect to be included: expansion of the universe during infla-tion. The size of the universe depends on time according to: U ( t ) = U e Ht , (2.16)where U is the initial volume of the pre-inflationary universe. One can interpret the proba-bility P r [ φ > φ c , t ] as fraction of the volume U inf ( t ) still inflating, that is: U inf ( t ) = U e Ht P r [ φ > φ c , t ] , (2.17)then in order for the Universe to inflate eternally, the positive exponential factor H inEq. (2.17) and the negative exponential factor − Γ in (2.14) must satisfy: H > Γ . (2.18)We shall illustrate this general property on an example of linear potential. Evaluating theintegral for probability density, in the linear case gives: P r [ φ > φ c , t ] = 12 erfc (cid:32) α H t − φ cH π √ Ht (cid:33) . (2.19)The error function may be approximated by an exponential: P r [ φ > φ c , t ] = C ( t ) exp (cid:18) − π α H t (cid:19) , (2.20)where C ( t ) is power-law in t and φ c vanished from the final approximation of the probability,which is a generic feature. By comparing the exponents we can check, whether U inf will growor tend to zero. The condition for eternal inflation to occur becomes: H > π α H . (2.21)– 5 –or linear potential α = V (cid:48) ( φ ) using the slow-roll equations equation (2.3), above conditioncan be rewritten: | V (cid:48) | V < √ π M P l . (2.22)This can be interpreted as quantum fluctuations dominating over classical field rolling. Forlinear potential, this is satisfied for a large φ . Similarly, the second condition for the eternalinflation may be derived from the quadratic hilltop potential: − V (cid:48)(cid:48) V < M P l . (2.23)Further necessary conditions on p-th derivative with p > have been derived in [1] and give: [ − sgn ( ∂ p V )] p +1 | ∂ p V | V (4 − p / < N p M p − P l , (2.24)where N p (cid:29) is numerically determined coefficient. In order to cross check the formulas(2.22, 2.23) the numerical simulation has been developed. To reconstruct the probabilitydistribution one simulates the discretized version of equation (2.6): φ n = φ n − − H V (cid:48) ( φ n − ) δt + δφ q ( δt ) , (2.25)with δφ q ( δt ) being random number taken from the gaussian distribution with mean equalzero, and variance H π δt . Given parameters, the field evolution is repeated many times toobtain the normalized probability density of the results, determining the possibility of theeternal inflation. In order to recognize the eternally inflating models we search for such initialvalue of the field φ that Γ < H . In the numerical analysis we eliminate φ c in equation(2.13) and instead check for the slow-roll conditions violation at each step. Most of the inflationary potentials are of the single-minimum type such as Starobinsky in-flation and alpha-attractors. There are however, potentials which are of type depicted onfigure 1 and possess a various minima. In such models, the inflation can become eternaldue to tunnelling to the false vacua. When the vacua are degenerate enough, the tunnelingdominates over quantum uphill rolling. The tunnelling goes in the opposing direction to theold inflation scenario [76] as showed on figure 1(b). As it will turn out this is the dominanteffect for the model discussed in section 4.3. The eternal inflation mechanism discussed in theprevious sections relies on the local shape of the potential and cannot provide an accuratedescription in that case. In order to quantitatively derive predictions for this new effect, weshall rely on the first passage formalism [84, 85] instead, and apply it to the eternal inflationconsiderations. Given the initial value of the field φ being between φ − and φ + , the probabil-ity that it reaches φ + before φ − and φ − before φ + , denoted respectively p + ( φ ) and p − ( φ ) ,obeys the following equation: vp (cid:48)(cid:48)± ( φ ) − v (cid:48) v p (cid:48)± ( φ ) = 0 , (2.26)– 6 – a) (b) Figure 1 . Left: field initially placed at the maximum of the potential may decay towards one of thetwo vacua: at φ − with probability p − and at φ + with probability p + .Right: field initially placed at φ < φ max may tunnel trough the barrier towards φ + with probability p + ( φ ) . Analogous tunneling from "plus" to "minus" side is also possible. with initial conditions: p ± ( φ ± ) = 1 , p ± ( φ ∓ ) = 0 , where v = v ( φ ) is the dimensionlesspotential: v ( φ ) := V ( φ )24 π M P l . (2.27)The analytical solution is: p ± ( φ ) = ± (cid:82) φ φ ± e − v ( φ ) dφ (cid:82) φ + φ − e − v ( φ ) dφ . (2.28)One may also define the probability ratio R : R ( φ ) := p + ( φ ) p − ( φ ) = (cid:82) φ − e − v ( φ ) dφ (cid:82) φ + φ e − v ( φ ) dφ . (2.29)The above integrals may be evaluated numerically. However, if the amplitude of v ( φ ) ismuch smaller than 1, the term e − /v ( φ ) will be extremely small, possibly exceeding machineprecision in the computation.Yet, one can use the steepest descent approximation. Consider a potential with the fieldlocated initially on the local maximum φ max with a minimum on each side, as depicted onfigure 1(a). Then the probability ratio R may be evaluated approximately, where the leadingcontributions to (2.29) come from the values of the field in the neighborhood of φ max . p + and p − gives a probability of evolution realised respectively by the red and the green ball.We get: R ( φ max ) ≈ − √ π v ( φ max ) v (cid:48)(cid:48)(cid:48) ( φ max ) | v (cid:48)(cid:48) ( φ max ) | / . (2.30) For the details of the calculation consult [85]. – 7 –n this regime, probability of descending into each of the minima φ − and φ + is similar, giving | − R | (cid:28) . It is possible to start the inflation in the subset of [ φ − , φ + ] that would leadto the violation of the slow-roll conditions and tunnel through the potential barrier to thesector dominated by eternal inflation, schematically shown on figure 1(b). We further analyzethis possibility in Sec. 4.3 for a particular effective potential with two vacua, stemming froman asymptotically safe theory. We use equation (2.30) to find the dependence of R onthe parameters of the theory and verify the result with direct numerical simulation of theLangevin Equation (2.6) given set of parameters. In this section we show the basic application of the conditions (2.22, 2.23) to simple effectivepotentials stemming from the α -attractor models and the Starobinsky inflation. We start our investigation with the α -attractor models [86], a general class of the inflationarymodels, originally introduced in the context of supergravity. They are consistent with theCMB data, and their preheating phase has been studied on a lattice in [87]. The phenomeno-logical features of these models are described by the Lagrangian: √− g L T = 12 R − ∂φ (cid:16) − φ α (cid:17) − V ( φ ) . (3.1)Here, φ is an inflaton and α can take any real, positive value. At the limit α −→ ∞ the scalarfield becomes canonically normalized, and the theory coincides with the chaotic inflation.Canonical and non-canonical fields are related by the transformation: φ = √ α tanh ϕ √ α . (3.2)We further consider T-models, in which the potential of canonically normalised field is givenby: V ( φ ) = αµ tanh n ϕ √ α , (3.3)where parameter µ is of order − . The shape of the potential for n = 1 was plotted on figure2(a). At the large φ the potential (3.3) is asymptotically flat, this creates the possibility foreternal inflation to occur. Using the first condition (2.22) we have verified, that generally thespace is eternally inflating for all initial values of the φ , above certain φ EI . The second eternalcondition (2.23) as well as the higher order conditions are satisfied for almost all values of φ above , providing no new information. This is a generic feature for all of the models weinvestigate.For every α , φ necessary to produce 60 e-folds is safely below φ EI . We have found φ bysolving slow-roll equation numerically. It is shown on figure 2(b). The time of inflation largerthan 60 Planck-times is unlikely according to the Planck Collaboration data [88]. The valuesof φ EI change only slightly with n . We may therefore conclude, that α -attractor models areconsistent with the beginning of "our" pocket universe. However, it is not inconceivable thatthe field fluctuations in other parts of the early universe had values φ > φ EI , driving theeternal inflation. – 8 – a) (b) Figure 2 . Left: the T-model potential with n = 1 and various α were depicted.Right: plot of the initial value φ necessary for 60 e-folds, as a function of α (blue), as well as thelowest initial value φ EI of the field, at which the eternal inflation "kicks in" (yellow). Solutions stemming from Einstein-Hilbert action predict initial singularity. In 1980 Starobin-sky proposed a model [89], where pure modified gravitational action can cause non-singularevolution of the universe, namely: S = 12 (cid:90) (cid:112) | g | d x (cid:18) M p R + 16 M R (cid:19) , (3.4)this can be rewritten to the effective potential form, with: V ( φ ) = V (cid:32) − exp (cid:32) − (cid:114) φM P l (cid:33)(cid:33) . (3.5)The inflation begins on a plateau at large φ . The field rolls towards a minimum at φ = 0 ,where the oscillatory reheating phase occurs. It has been estimated from the CMB data,that during the inflation the volume of the universe has grown by approximately 60 e-folds.This corresponds to the initial condition φ = 5 . M P l , without taking into account quantumgravity effects. It is possible to perturbatively recover information about the shape of thepotential from the CMB, for details see [90]. Amplitude of the scalar power spectrum A s =2 × − fixes the value V = 8 . × − M P l , via the relation: V = 24 π (cid:15) ( φ ) A s . (3.6)Applying the analytical eternal inflation conditions (2.22, 2.23) to the Starobinsky potential,the initial value of the field, above which the eternal inflation occurs, has been estimated to be φ = 16 . M P l . It has been found, that decay rate Γ decreases approximately exponentiallywith φ . Our numerical simulation confirms the analytical prediction discussed in [1] withina sample of 10000 simulations. Nevertheless, for eternal inflation scenario and realistic phe-nomenology, this model requires the transplanckian values of the fields in order to reproducethe correct tensor to scalar ratio r , amplitudes and spectral tilt n s . Hence the Starobinskyinflation will be affected, possibly invalidated by the quantum gravity fluctuations. The lead-ing log corrections have been studied in [31], which we as well study in the context of eternal– 9 – igure 3 . Left: exemplary field evolution for the Starobinsky model has been plotted. Green plotshows the solution to the classical slow-roll equation, and the black plot is the Langevin solution. Theinflation ends, when the slow-roll parameter reaches 1. Values of the field are given in M P l .Right: the time dependence of the probability, that inflation still occurs. In the slow-roll regime, theprobability decays exponentially. Linear fit slope, the decay rate is around
Γ = 0 . . Red, dashedline denotes eternal inflation threshold with slope H of order − . Since H < Γ , initial condition φ = 3 M P l is an example of non-eternally inflating universe. inflation. Yet, due to the large field values required for eternal inflation to occur, one shouldseek a theory predictive up to an arbitrary large energy scale, which we discuss in the nextsection.
In this chapter, as a warm up we study effective corrections to the Starobinsky inflation,providing a different behavior at the large field values. Later we show, that RG-improvementof Starobinsky model proposed in [91], see also [92, 93] for review, closely related to the R + R renormalizable Fradkin-Tseytlin gravity [94] produces a branch of inflationary potentialentirely dominated by eternal inflation. We find the initial values of the inflaton such thatinflation becomes eternal, for the remaining branch, as a function of theory parameters.Finally we show, that the possibility of tunneling through the potential barrier present in [95]becomes a new mechanism for eternal inflation. In all of the asymptotically safe inflationarytheories, eternal inflation is present as a consequence of asymptotic flatness of the effectivepotential. Below Planck scale the gravitational constant G N has a vanishing anomalous dimension andthe R has a coefficient that runs logarithmically [96] (this comes from the fact that R isdimensionless in dimensions). Hence, one can motivate various quantum corrected infla-tionary models, such as [28–31]. In particular, the leading-log corrections to the Starobinskymodel are given by [31]: L eff = M P l R a R b ln (cid:16) Rµ (cid:17) + O ( R ) . (4.1)– 10 –n order to find the Einstein frame potential for this model a few steps need to be taken.First, we use the conformal transformation [97]. Then, following the next transformation forthe Ricci scalar and the metric determinant we get the Einstein frame action: S = (cid:90) d x √− g E (cid:20) M P l R E − g µνE ( ∂ µ φ E ) ( ∂ ν φ E ) − V E ( φ E ) (cid:21) , (4.2)which depends on the sought potential V E , that can be further obtained: V E (Φ) = M P l a Φ (cid:16) b ln (cid:16) Φ µ (cid:17)(cid:17) (cid:16) b ln (cid:16) Φ eµ (cid:17)(cid:17)(cid:26) M p (cid:16) b ln (cid:16) Φ µ (cid:17)(cid:17) + 2 a Φ (cid:16) b ln (cid:16) Φ √ eµ (cid:17)(cid:17)(cid:27) , (4.3)with φ E given by F ( φ E ) = M P l exp (cid:32)(cid:114) φ E M P l (cid:33) = M P + a Φ[2 − b + 2 b ln (cid:0) Φ /µ (cid:1) ][1 + 2 b ln(Φ /µ )] , (4.4)yet the transformation between Φ and φ E is non-invertible. By taking into account the COBEnormalization, we can treat b as a free parameter and fix a ( b ) . For b = 0 one obtains the usualStarobinsky model, and for b (cid:28) one gets the model discussed in [29] R (1 + β ln R ) withthe potential given by Lambert W function (so the same as for model discussed in section4.3) and approximated in the limit β (cid:28) as V ≈ V s b/ (2 α ) + β/α ln[( e ˜ χ − / α ] , (4.5)where V s is the Starobinsky potential, ˜ χ is the Einstein frame field and α ( β ) , where we havekept the original notation.From the plots 4(a), 4(b), one can see that both of the models should give similar inflationaryobservables as in the Starobinsky inflation. On the other hand the eternal inflation in thismodels will be quite different. These models for β < and b > have the potentials thatare non-flat for large field values, while for β > potential depicted on Figure 4(b) have therunaway behaviour, so different asymptotic behaviour, which is discussed in Section 4.3. Thismakes those potentials quantitatively different from the Starobinsky model in the context ofeternal inflation and suggest, that eternal inflation cannot take place in those models. Tobe concrete, we have checked that eternal inflation for model described by 4.3 takes place at Φ ≈ M P l , which is far beyond the applicability of the model. So now we turn to theinflationary models stemming from the asymptotic safety. Here, the effective action differs from [31] due to the introduction of auxiliary numerical parameter e ≈ . . Nevertheless, the dynamics stemming from each of the potentials are equivalent. – 11 – a) (b) Figure 4 . Left: Plots of the potential (4.3) for various b parameters.Right: Potential (4.5) in dependence of β parameter values. Approach towards infinity in case of β > is visible. Renormalization Group improvement is a procedure of identifying and replacing the RG scale k with a physical scale. It incorporates leading-order quantum effects in the dynamics ofclassical system. In the case of gravity, running of coupling constants in Einstein-Hilbertaction results in additional contribution to the field equations from the gravitational energy-momentum tensor [93]. In the de Sitter-type setting k ∼ R is the unique identification of thephysical scale dictated by Bianchi identities [93]. Such replacement in the scale-dependentEinstein-Hilbert action generates an effective f ( R ) action, whose analytical expression isdetermined by running of the gravitational couplings. RG-improvement could solve classicalblack hole singularity problem [98, 99], gives finite entanglement entropy [100] and generatesinflationary regime in quantum gravity [37].In this section, we study the asymptotically safe inflation based on RG-improved quadraticgravity Lagrangian, considered in [91, 93]: L k = 116 πg k (cid:0) R − λ k k (cid:1) − β k R , (4.6)with the running dimensionless couplings g k , λ k , β k being the three relevant directions of thetheory, with running given by [101]: g k = 6 πc k πµ + 23 c ( k − µ ) , β k = β ∗ + b (cid:18) k µ (cid:19) − θ / , (4.7)where µ is the infrared renormalization point such that c = g k ( k = µ ) and c and b areintegration constants. We introduce a parameter α as α = − µ θ b /M P , (4.8)that measures the departure from the non-gaussian fixed point (NGFP). One may find thebehavior of the couplings near the NGFP and substitute the appropriate expressions to theLagrangian using the RG-improvement and the following identification of scale k = ξR ,where ξ is an arbitrary parameter of order one. Following [91] we shall assume θ = 1 , then– 12 – a) (b) Figure 5 . Left: V + ( φ ) plot for various α and fixed Λ = 1 is shown.Right, logarithmic dependence of initial field value above which eternal inflation occurs on parameter α has been found. Blue points were evaluated via (2.22). the transformation from the Jordan to the Einstein frame yields an effective potential [91, 93]: V ± = m e − (cid:113) κφ κ (cid:40) e (cid:113) κφ − − α + 128Λ − √ α (cid:34) ( α + 8 e (cid:113) κφ − ± α (cid:113) α + 16 e (cid:113) κφ − (cid:35) − α ( α + 16 e (cid:113) κφ − ∓ α (cid:113) α + 16 e (cid:113) κφ − (cid:41) , (4.9)where the only free parameters are cosmological constant Λ and α after the CMB normal-ization we perform below. The V + branch predicts the reheating phase, figure 5(a) shows itsplot for various α .Similarly as in the case of Starobinsky inflation, we have denoted V the constant part of thepotential at infinity V ( φ −→ ∞ ) = V = m κ and fixed it with CMB data by the relation (3.6).For example, given α = 2 . , Λ = 1 the plateau value is equal to V = 1 . × − M P l , henceone may fix the mass parameter m = 2 × GeV.Now we shall investigate the eternal inflation conditions given by (2.22, 2.23). These condi-tions restricts the initial value of the field. We search for φ above which the eternal inflationoccurs, as a function of the theory parameters. We have also found, that initial value abovewhich eternal inflation occurs does not depend on the cosmological constant. It is due tothe fact that Λ only shifts the minimum of the potential and does not affect the large-fieldbehavior of the system. Analytical conditions for EI have been checked for a set of α anddepicted on figure 5(b). The initial value of the field, depends logarithmically on α . Thereason for that behaviour is the following. In the large field expansion: V ± ( φ ) = V plateau − V αe − (cid:113) φ , (4.10)and by the substitution ˜ φ = e − (cid:113) φ the potential reduces to the linear hilltop model, whichjustifies the usage of formulae (2.22, 2.23) and the functional form of φ ( α ) . The results– 13 –ere also confirmed by the numerical simulations. For example, given Λ = 1 , α = 1 . ,the analytical considerations predict φ EI = 22 . M P l . The direct numerical simulation forthis set of parameters yields
Γ = 0 . M P l , and H = 0 . M P l , meaning that theeternal inflation begins slightly below the expected value φ EI . The plateau of (4.9) at largefield values is a characteristic feature of effective inflationary potentials stemming from theasymptotically safe theories. It is dominated by eternal inflation and may suggest a deeperrelation between the asymptotic safety of quantum gravity and multiverse. In this section we investigate model in which inflation is driven by an ultraviolet safe and in-teracting scalar sector stemming from a new class of non-supersymmetric gauge field theories.We consider a
SU( N C ) gauge theory, with N F Dirac fermions and interacting with an N F × N F complex scalar matrix H ij that self interacts, described in [95]. The Veneziano limit( N F → + ∞ , N C → + ∞ , N F /N C = const ) is taken such that the ratio N F /N C becomes acontinuous parameter [70]. The action in Jordan frame has the following form: S J = (cid:90) d x √− g (cid:26) − M + ξφ R + g µν ∂ µ φ∂ ν φ − V iUVFP (cid:27) , (4.11)where the leading logarithmically resummed potential V iUV F P is given by: V iUVFP ( φ ) = λ ∗ φ N f (1 + W ( φ )) (cid:18) W ( φ ) W ( µ ) (cid:19) δ , (4.12)where λ ∗ = δ π ( (cid:112)
20 + 6 √ − √ − is positive quartic coupling at the fixed point, φ is the real scalar field along the diagonal of H ij = φδ ij / (cid:112) N f and δ = N F /N C − / is thepositive control parameter, W ( φ ) is the Lambert function solving the transcendent equation z = W exp W, (4.13)with z ( µ ) = (cid:18) µ µ (cid:19) δα ∗ (cid:18) α ∗ α − (cid:19) exp (cid:20) α ∗ α − (cid:21) . (4.14)The parameter α ∗ = δ + O ( δ ) is the gauge coupling at its UV fixed point value and α = α ( µ ) is the same coupling at a reference scale µ .A conformal transformations allows to rewrite the action from Jordan to Einstein frame.Assuming a single field slow-roll inflation, we examine inflationary predictions of the potentialand compute the slow-roll parameters: (cid:15) = M P l (cid:18) dU/dχU (cid:19) , η = M P l d U/dχ U , (4.15)where U = V iUVFP / Ω , with Ω = ( M + ξφ ) /M P l being the conformal transformation ofthe metric, and χ is the canonically normalized field in the Einstein frame. We assume, that M = M P l . Inflation ends when the slow-roll conditions are violated, that is when (cid:15) ( φ end ) or| η ( φ end ) | = 1. We analyze the non-minimal case, where the coupling ξ is non-vanishing. Thepotential U is given by: U = V iUVFP Ω ≈ λ ∗ φ N F (cid:16) ξφ M Pl (cid:17) (cid:18) φµ (cid:19) − δ . (4.16)– 14 – a) (b) Figure 6 . Left: the non-minimally coupled potential as a function of φ for δ = 0 . , ξ = 1 / and µ = 10 − M P l . There is a maximum at φ max = 16 . M P l .Right: For the same set of parameters, we plot the first eternal inflation condition as a function of φ (blue curve) and the eternal inflation bound (yellow curve). Inflation becomes eternal if the blue curveis below the yellow one. At φ max the first derivative of the potential vanishes and (2.22) predicts anarrow window for the eternal inflation. In the large field limit φ (cid:29) M P l / √ ξ the φ term in the numerator cancels against theterm in the denominator. In this limit, the quantum corrections dictate the behaviour of thepotential, which is found to decrease as: λ ∗ M P l N F ξ (cid:18) φµ (cid:19) − δ . (4.17)The non-minimal coupled potential has one local maximum and two minima. The region tothe left of the maximum is the region, where the inflation can be brought to an end and thereheating takes place [102]. To the right of the maximum, the inflation becomes classicallyeternal. For large values of φ the potential flattens out and the slow-roll conditions are notviolated. Numerical solutions to the FP equation shows that the is no possibility of eternalinflation in that region, since it is an unstable maximum and hence any quantum fluctuationwill drop it from that position. Furthermore due to steepness of the potential around thismaxima there is no possibility for the field to remain in that region. Let us now investigatethe analytical eternal inflation conditions. Similarly as in the Starobinsky model, the secondcondition (2.23) is always satisfied. The first condition (2.22) is illustrated on the figure 6(b).There is a peak for φ = φ max = 16.7 M P l , due to the vanishing derivative and if we "zoom in",the analytical condition allows for eternal inflation in the close neighbourhood of φ max . Wehave verified numerically this is not a sustainable attractor of eternal inflation. A field thatstarts evolution at φ max will leave this region, as it cannot climb further up-hill. Nevertheless,eternal inflation may still occur due to the quantum tunnelling through the potential barrier. Tunneling through the potential barrier
As described in section 2.4, if the potentialhas multiple vacua, quantum tunneling through the potential barrier is expected. The non-minimal coupling potential (4.16) belongs in this class. The question is, whether tunnellingfrom the non-eternal inflation region of φ < φ max to the region of classical eternal inflation φ > φ max is possible.We start with investigating the fate of the field initially placed at the peak φ = φ max of– 15 – a) (b) Figure 7 . Left: probability ratio R = p + p − of descending from the maximum towards the rightminimum ( p + ) and the left minimum ( p − ), as a function of theory parameters ξ and δ . For the smallvalues of the parameters it is more probable to fall from the maximum towards φ − = 0 with non-eternal inflation, while for the large values of parameters, minimum at φ + = ∞ is favored, resultingin eternally inflating universe.Right: the value of the field at which, the potential is maximal. The above figures are qualitativelysimilar because for the small values of φ max , the effective potential is highly asymmetric ("step-like").This brakes the symmetry between the right and left descend probability. the potential depicted on figure 6(a). By the virtue of the steepest descent approximation atthe maximum equation (2.30) may be employed. The resulting ratio of probabilities of theright-side descent to the left-side descent R ( α, ξ ) = p + p − , as a function of parameter α and thenon-minimal coupling constant ξ is shown on figure 7(a). As expected, the ratio is close to 1and favors the right-side (left-side) of the potential for large (small) values of the parameters.The biggest ratio emerges at large values of the parameters ξ and δ , since the potential is"step-like" and highly asymmetric. It is monotonically decreasing with the values of φ max ,for which the potential has a maximum, see figure 8(b).In order to verify the accuracy of the relation (2.30), we have performed numerical sim-ulation of the discretized Langevin equation (2.25) with initial condition φ = φ max . Forexample it was found, that for the set of the parameters N F = 10 , µ = 10 − M P l , ξ = , δ = 0 . the steepest descent approximation yields R = 0 . , and the numerical analysis re-sults in R = 0 . , which proves good accuracy of the analytical formula.One may wonder, how does the probability p ± ( φ ) depends on the departure from the maxi-mum φ (cid:54) = φ max . The analytical answer is given by (2.28). As we have checked numericallyinflation becomes eternal, when tunneling probability is non-zero, as depicted on Figure 8(a).To bypass the numerical calculation of the integral (2.28) we employ the direct numericalsimulation of the Langevin equation. As expected, choosing values of φ smaller than φ max lowers the probability of the tunneling to the right side of the barrier. Moreover, the prob-ability of tunneling decreases linearly with the distance to the maximum. The result of the– 16 – a) (b) Figure 8 . Left: Linear probability distribution of tunneling (green side) and rolling (red side) towardsthe minimum at infinity as. The data points have been directly simulated.Right: Probability ratio R , evaluated with (2.30), is a monotonically decreasing with the value of themaximum of the potential in the steepest descent approximation simulation for the set of parameters N F = 10 , µ = 10 − M P l , ξ = 16 , δ = 0 . (4.18)is shown on figure 8(a). The green line corresponds to the green ball on figure 1(a) and showsthe probability of tunneling through the barrier (as in figure 1(b)) as a function of proximityto the maximum φ (cid:54) = φ max . The red line corresponds to the red ball on figure 1(a) andshows the probability of rolling towards the minimum at infinity.The rolling is also a stochastic process, as the tunneling in the opposite direction is possi-ble. The probability distribution of tunneling in either direction is not a symmetric process.Notice, that the initial condition, for which p + = is shifted to the right of φ max . Thismeans, that starting from the maximum, it is slightly more probable to land in φ − . There isa point, below which the green ball cannot tunnel p + = 0 (for the set of parameters given by(4.18), at . M P l ), and the other limiting case (at . M P ), when the red ball cannot tunnel p − = 0 . Hence, for every initial value of the field above . Planck Masses, there is a non-zeroprobability of eternal inflation. On the other hand, for φ = 9 . M P l and parameters given by(4.18), the inflation classically produces roughly 54 e-folds, depending on the reheating time[102], and is in the agreement with CMB data. This shows that the model is on a verge ofbeing eternally inflating, which may point out to the interesting phenomenology.To sum up the critical point of our analysis is that the analytical conditions (2.22, 2.23) didnot allow for eternal inflation, even though the tunneling process may evolve any initial pointabove 9.2 Planck Masses to φ + = ∞ , and not violate the slow-roll conditions. This shows thatconditions (2.22,2.23) are not well suited for the multiple minima models and cannot containthe full information of the global influence of quantum fluctuations in the early universe. Eternal inflation remains a conceptual issue of the inflationary paradigm. The creation ofscattered, causally disconnected regions of spacetime - the multiverse is not confirmed obser-vationally and raises question about the inflationary predictions [77]. Hence, one may impose– 17 –he no eternal inflation principle [1] to restrict free parameters and the initial conditions.We have investigated popular inflationary models, and have found that in principle, eter-nal inflation is present at every asymptotically flat effective potential for large field values,assuming the ergodicity of the system. The finite inflationary time of our pocket universeserves as a consistency condition of the multiverse predictions. In section 3.1, we verified that α -attractor T-models are consistent from this point of view.If the initial value of the scalar field driving inflation is above the Planck scale, a UV-completeness of a given model is necessary. Starobinsky inflation stemming from R gravitygives around 60 e-folds for φ =5.5 M pl . We have considered the effective quantum correc-tions to the Starobinsky inflation based on the qualitative behavior of the running couplingconstants. Next, the RG-improvement of R + R Lagrangian was studied in this case andwe have found that field values required for eternal inflation are typically higher than theones for the Starobinsky case. The flatness of the potential and possibility of eternal inflationseems to be a signature mark of asymptotically safe UV completions, in contradistinction tothe effective theory corrections. We have checked that [31] Φ ∼ M P l in order to geteternal inflation, which is far beyond the applicability of the model.Furthermore we have found, that for potentials with multiple vacua, tunneling through po-tential barriers provides a new mechanism for eternal inflation. So in order to understandthe inflationary dynamics one cannot simply cut the potential at the maximum. The
SU( N ) Gauge theory with Dirac fermions provides an example for such behavior. The probabilityof tunneling to the side dominated by eternal inflation becomes negligible few Planck Massesaway from the peak of the potential. Yet the fixed point values of the couplings and possiblythe shape of the potential can be obscured by the quantum gravity effects and this shall beinvestigated elsewhere.Our analysis reveals that there is no obstruction for the multiverse scenario in the asymptot-ically safe models. Yet its occurrence depends on the initial conditions for the inflationaryphase and the matching to the observational data, tying these three profound issues together.On the other hand, in AS models these questions can have intriguing answers by the finiteaction principle [78].
Acknowledgments
We thank J. Reszke and J. Łukasik for participating in the early stages of this project.We thank G. Dvali, A. Eichhorn, M. Pauli, A. Platania, T. Rudelius, S. Vagnozzi and Z.W.Wang for fruitful discussions and extensive comments on the manuscript. Work of J.H.K. wassupported by the Polish National Science Center (NCN) grant 2018/29/N/ST2/01743. J.H.K.would like to acknowledge the CP3-Origins hospitality during this work. The computationalpart of this research has been partially supported by the PL-Grid Infrastructure.
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