Negative running prevents eternal inflation
NNegative running prevents eternal inflation
William H. Kinney ∗ Dept. of Physics, University at Buffalo, the State University of New York, Buffalo, NY 14260-1500
Katherine Freese † Department of Physics, University of Michigan, Ann Arbor, MI 48109 (Dated: October 16, 2018)Current data from the Planck satellite and the BICEP2 telescope favor, at around the 2 σ level,negative running of the spectral index of curvature perturbations from inflation. We show thatfor negative running α <
0, the curvature perturbation amplitude has a maximum on scales largerthan our current horizon size. A condition for the absence of eternal inflation is that the curvatureperturbation amplitude always remain below unity on superhorizon scales. For current bounds on n S from Planck, this corresponds to an upper bound of the running α < − × − , so that eventiny running of the scalar spectral index is sufficient to prevent eternal inflation from occurring, aslong as the running remains negative on scales outside the horizon. In single-field inflation models,negative running is associated with a finite duration of inflation: we show that eternal inflation maynot occur even in cases where inflation lasts as long as 10 e-folds. I. INTRODUCTION
Inflation [1–3] has emerged as the standard paradigmfor modeling the behavior of the very early universe. Inaddition to explaining the flatness and homogeneity ofthe cosmos, inflation predicts the generation of pertur-bations from quantum fluctuations in the early universe[4–15], a prediction which has been tested to high pre-cision in measurements of the Cosmic Microwave Back-ground (CMB). The temperature anisotropy of the CMBhas been measured in exquisite detail by the Planck satel-lite [16–18], and recent measurement of the CMB polar-ization by the BICEP2 telescope has provided clear ev-idence of primordial gravitational waves consistent withthe predictions of inflation [19].The Planck and BICEP data are consistent with thesimplest inflationary models. An example is inflation ina quadratic monomial potential, V ( φ ) ∝ φ , which pre-dicts a tensor/scalar ratio r (cid:39) .
15, consistent with BI-CEP2 constraints, and a scalar spectral index n S (cid:39) . φ > M P duringinflation, where M P is the reduced Planck Mass. Suchpotentials have the interesting property that, for fieldvalues φ (cid:29) M P , the amplitude of quantum fluctuationsin the field becomes larger than the classical field varia-tion, so that the field is as likely to roll up the potentialas it is to roll down the potential. Therefore, in a sta-tistical sense, inflation never ends: there will always beregions of the universe where the field has fluctuated up-ward, rather than downward, and inflation becomes aquasi-stationary, infinitely self-reproducing state of eter-nal inflation [20–23].In this paper we consider the question of whether the ∗ Electronic address: whkinney@buffalo.edu † [email protected] BICEP2 data imply eternal inflation. We focus in partic-ular on the fact that Planck + BICEP2 weakly favor ascale-dependent spectral index, a so-called running of theprimordial power spectrum. Such running of the powerspectrum would rule out all simple monomial potentials,requiring a more complex (and more finely tuned) infla-tionary potential. We find that for even a small nega-tive running, of order α ∼ − , eternal inflation is pre-vented, even in cases where inflation continues for manye-folds. Therefore, it is premature to conclude that thelarge tensor signal favored by BICEP2 is consistent onlywith models leading to eternal inflation. The paper isorganized as follows: Section II considers the current ev-idence for running of the spectral index. Section III dis-cusses the relationship between eternal inflation and theamplitude of the curvature perturbation spectrum. Sec-tion IV discusses suppression of eternal inflation in thecase of negative running of the curvature power spec-trum. Section V presents a summary and conclusions. II. EVIDENCE FOR RUNNING OF THESPECTRAL INDEX
While a simple monomial potential leading to eternalinflation is consistent with the Planck and BICEP2 data,there is tension at approximately the 2 σ level betweenthe Planck and BICEP2 constraints on the amplitude oftensor perturbations [24]. The tension arises from theanomalously low signal in the temperature anisotropy asobserved by Planck on large angular scales [25]. TheBICEP2 constraint on the tensor amplitude exacerbatesthe anomaly, since tensor modes also contribute to thetemperature anisotropy on exactly the scales where thePlanck CMB power is anomalously low: If, as suggestedby BICEP2, as much as 20% of the large-scale tempera-ture anisotropy is from tensors, this requires even moredrastic suppression of curvature perturbations than inthe zero-tensor case. This tension can be alleviated by a r X i v : . [ a s t r o - ph . C O ] J un assuming that the shape of the curvature power spectrumis itself scale-dependent, i.e. by including running of thepower spectrum, α ≡ dn S d ln k , (1)where n S is the spectral index of curvature perturbations, n S ≡ d ln P ( k ) d ln k . (2)Other possibilities for resolving the tension include an ex-tra neutrino species [26–29] features in the tensor powerspectrum [30, 31], early Dark Energy [32], a non-Bunch-Davies initial state [33, 34], isocurvature perturbations[35], or a rapid phase transition during inflation [31].Here we focus on the possibility of running. If runningis included in a fit to the Planck + BICEP2 data, it isfavored at approximately the 95% confidence level [36–41]. Figures 1 and 2 show joint constraints on the ten-sor/scalar ratio r , the scalar spectral index n S , and therunning α for Planck and BICEP2. The constraints aregenerated using the COSMOMC Markov Chain MonteCarlo code [42], marginalizing over a eight-parameterdata set with flat priors: • Dark Matter density Ω M h . • Baryon density Ω b h . • Reionization optical depth τ . • The angular size θ of the sound horizon at decou-pling. • Scalar spectrum normalization A S . • Tensor/scalar ratio r . • Scalar spectral index n S . • Running α .The fit assumes a flat universe Ω b + Ω M + Ω Λ = 1, withCosmological Constant Dark Energy, ρ Λ = const . Con-vergence is determined via a Gelman and Rubin statistic.Auxiliary data sets used are WMAP polarization (WP),in combination with the Atacama Cosmology Telescope(ACT) / South Pole Telescope (SPT) CMB measure-ments (solid contours in figures), and Baryon AcousticOscillation (BAO) data from Sloan Digital Sky SurveyData Release 9 [43], the 6dF Galaxy Survey [44], andthe WiggleZ Dark Energy Survey [45] (dashed contours).The pivot scale is taken to be k (cid:63) = 0 . h MpC − .We see that a running power spectrum is favored atroughly 95% confidence, a result which can be consideredsignificant from the standpoint of Bayesian evidence [46].In this paper, we consider the significance of this resultfor the hypothesis of eternal inflation. FIG. 1: Constraints on the tensor/scalar ratio r and running α . Solid contours show constraints from Planck + WMAPPolarization + Lensing + ACT + SPT + BICEP2. Dashedcontours show constraints from Planck + WMAP Polariza-tion + Lensing + BAO + BICEP2. The pivot scale is k (cid:63) = 0 . h MpC − .FIG. 2: Constraints on the scalar spectral index n S andrunning α . Solid contours show constraints from Planck +WMAP Polarization + Lensing + ACT + SPT + BICEP2.Dashed contours show constraints from Planck + WMAP Po-larization + Lensing + BAO + BICEP2. The pivot scale is k (cid:63) = 0 . h MpC − . III. ETERNAL INFLATION AND THECURVATURE PERTURBATION
So-called “eternal” inflation occurs when quantumfluctuations in the inflaton field dominate over the classi-cal field evolution, where the amplitude of quantum fluc-tuations in the inflaton field during inflation is δφ Q ≡ (cid:10) δφ (cid:11) / = H π . (3)For eternal inflation to occur, this quantum fluctuationamplitude must be larger than the classical field variationover approximately a Hubble time, δφ C = ˙ φH . (4)Therefore, the condition for eternal inflation can be writ-ten δφ Q δφ C = H π ˙ φ > . (5)We note that the fraction (5) is identical to the ampli-tude of the curvature perturbation for modes crossing thehorizon during inflation, P ( k ) = H π ˙ φ . (6)The curvature perturbation is simply the amplitude ofquantum fluctuations in the inflaton in units of the fieldvariation in a Hubble time! Accordingly, the conditionfor eternal inflation is that the curvature perturbationamplitude must exceed unity [47, 48], P ( k ) > . (7)In the next section, we show that for sufficiently negativerunning of the scalar spectral index, the scalar perturba-tion amplitude always remains below unity and eternalinflation never occurs. IV. AN UPPER BOUND ON RUNNING
The power spectrum for curvature perturbations canbe written in terms of the scalar spectral index n S andthe running α as P ( k ) = P (cid:63) (cid:18) kk (cid:63) (cid:19) n S − α ln( k/k (cid:63) )+ ··· , (8)where k (cid:63) is a pivot scale, which we take to be a scalecomparable to CMB observations, k (cid:63) = 0 . h MpC − , sothat a Planck + BICEP + BAO limit gives P (cid:63) = 2 × − .First consider the case of no running, α = 0, so that P ( k ) = P (cid:63) (cid:18) kk (cid:63) (cid:19) n S − . (9)The Planck satellite measurement of the CMB indicatesthat the spectral index is red , i.e. n S − <
0, with a95% confidence limit of approximately 0 . < n S < . k (cid:28) k (cid:63) , of P ( k ) > P (cid:63) (cid:18) k (cid:63) k (cid:19) . , k < k (cid:63) . (10) For a constant red spectral index, we then see that eter-nal inflation is inevitable in the limit k →
0, with P ( k )exceeding unity when kk (cid:63) < P / . (cid:63) (cid:39) − , (11)or roughly N (cid:39) − ln 10 − (cid:39) n S − < − .
02, eternal inflation is guaranteed as long as inflationcontinues for at least 1000 e-folds.We now consider the case of constant running, P ( k ) = P (cid:63) (cid:18) kk (cid:63) (cid:19) n S − α ln( k/k (cid:63) ) . (12)The Planck + BICEP limit on running of the spectralindex is approximately 0 ≥ α > − .
05, so that negativerunning is favored, and positive running is inconsistentwith the data at 95% confidence. Note that negativerunning means the spectral index gets redder on smallscales k → ∞ , and bluer on large scales, k →
0. Forconstant negative running α <
0, the spectral index for k (cid:28) k (cid:63) will eventually exceed unity, n − >
0. We nowshow that this is sufficient to prevent eternal inflation aslong as α is sufficiently negative. If eternal inflation is tobe evaded, this implies an upper bound on the curvatureperturbation spectrum P ( k ) < P ( k ) = ln P (cid:63) + (cid:20) n S − α ln (cid:18) kk (cid:63) (cid:19)(cid:21) ln (cid:18) kk (cid:63) (cid:19) < , (13)for all k < k (cid:63) . The curvature power spectrum will havean extremum at d ln P ( k ) d ln k = n S − α ln (cid:18) kk (cid:63) (cid:19) = 0 . (14)Solving for the wavenumber k givesln (cid:18) k max k (cid:63) (cid:19) = 1 − n S α . (15)This extremum is guaranteed to be a maximum as longas the running α is negative, since d ln P ( k ) d (ln k ) = α < . (16)For eternal inflation to be prevented, it is then sufficientthat the maximum of the curvature power spectrum beless than unity, orln P ( k max ) = ln P (cid:63) − (1 − n S ) α < . (17)This is equivalent to an upper bound on the running α of α < (1 − n S ) P (cid:63) . (18)From the CMB limits P (cid:63) (cid:39) × − and 1 − n S < . α < − × − . (19)For running below this upper bound, the curvature per-turbations amplitude remains smaller than unity for allwavenumbers k . Thus, even a very weak negative run-ning, consistent with slow-roll inflation, is sufficient toprevent eternal inflation from ever occurring.We have assumed that the running of the spectral in-dex is constant, i.e. that there is no running-of-running,running-of-running-of-running, and so on. There is, how-ever, no guarantee that the contributions from higher-order terms in the series( n S −
1) + α ln (cid:18) kk (cid:63) (cid:19) + β (cid:20) ln (cid:18) kk (cid:63) (cid:19)(cid:21) + · · · (20)do not become important in the limit k →
0. It is ev-ident that, as long as the higher-order terms are them-selves negative, eternal inflation will still never occur: α need not be constant to suppress eternal inflation, it mustsimply be negative. In addition, negative running of thespectral index also implies an eventual breakdown in slowroll at very large scales, indicating a finite total durationof inflation. We can estimate the total number of e-foldsof inflation by noting that dN = d ln a (cid:39) d ln k , where a ( t ) is the scale factor. Then, a breakdown of slow rollwill occur when (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dn S d ln k d ln k (cid:12)(cid:12)(cid:12)(cid:12) ∼ , (21)which gives N ∼ | ∆ ln k | ∼ / | α | . For running α ∼− .
05, near the outer limit of the Planck + BICEP al-lowed region, this means a very rapid breakdown of slowroll, in about 20 e-folds of inflation. However, for moremoderate running, a lengthy period of inflation is stillpossible, of order 10 e-folds for α ∼ − − . Inflationcan continue for an extended period without the onset ofeternal inflation. However, inflation in such cases is stillof finite duration. To avoid eternal inflation, it is suffi-cient that higher-order terms be subdominant during the finite period of inflation, for example α > β | ∆ ln k | , (22)or β < α . (23)This is a somewhat stronger condition than the assump-tion of slow roll, since in the slow roll approximation,the spectral index is first-order in slow roll parameters, n − ∼ O ( (cid:15), η ), α ∼ O (cid:0) (cid:15) , (cid:15)η, . . . (cid:1) , β ∼ O (cid:0) (cid:15) , . . . (cid:1) . Thiscondition is sufficient, but not necessary: eternal infla-tion may still be suppressed even if the running becomespositive, as long as the curvature perturbation remainsbelow unity. What does the lower bound (19) imply about the formof single-field inflationary potentials? In terms of slowroll parameters, we can write n S − η − (cid:15), (24)and α = − ξ + 16 (cid:15)η − (cid:15) , (25)where (cid:15) = M (cid:18) V (cid:48) V (cid:19) η = M (cid:18) V (cid:48)(cid:48) V (cid:19) ξ = M (cid:18) V (cid:48) V (cid:48)(cid:48)(cid:48) V (cid:19) . (26)For eternal inflation to be suppressed, the spectral indexmust change from red ( n <
1) on small scales to blue( n >
1) on large scales. Since (cid:15) is positive-definite, ablue spectrum means that η must be positive and largerelative to (cid:15) , η > (cid:15). (27)Taking V (cid:48) , V (cid:48)(cid:48) >
0, so that the potential becomes largefor large field values, we must then have dηdφ = M V (cid:48)(cid:48)(cid:48) V − M V (cid:48) V (cid:48)(cid:48) V > . (28)Therefore, negative running requires a large positivethird derivative V (cid:48)(cid:48)(cid:48) of the potential, or equivalently aslow-roll parameter ξ > (cid:15)η . A simple example of such apotential is inflation near an inflection point [49], V ( φ ) = V + Λ φ − m φ + µφ + · · · , (29)where the constants Λ, m , and µ all have dimensions ofmass. Near the inflection point φ = 0, η = M P µφ − m V , (30)which is positive for φ > m / µ , and negative for φ < m / µ , so the spectral index evolves from blue ( n S > n S <
1) as the field rolls down the potential. Asufficiently large tensor/scalar ratio can be generated bytuning of the coefficient of the linear term, since near φ = 0, r = 16 (cid:15) (cid:39) (cid:18) M P Λ V (cid:19) . (31)Such inflection point models have been suggested as typ-ical in the string landscape [50–56], albeit typically withvery small tensor/scalar ratios arising from the neces-sity of small field variation on the compactified manifoldstypical in string theory. A thorough dynamical analysisof inflection-point inflation can be found in Ref. [57].Similar models have been proposed from a strictly phe-nomenological viewpoint, with suppression of the curva-ture perturbation on large scales arising from a period offast-roll scalar field evolution [58–61]. Other possibilitiesinclude an early superinflationary phase [62], an earlynon-inflationary phase [63–67], double inflation [68], acurvaton [69], punctuated inflation [70, 71], or other newphysics [72, 73]. The possibilites for model-building ex-tend well beyond the simple potential (29). V. CONCLUSIONS
In this paper, we consider the viability of eternal infla-tion in light of the results from the Planck and BICEP2observations of the Cosmic Microwave Background. Cur-rent data weakly favor nonzero running of the scalar spec-tral index n S , mostly as a result of the suppressed scalarpower observed on large angular scales in the Planckdata. The suppression of low- (cid:96) modes in the CMB, com-pared to expectations from the standard Λ-CDM cosmol-ogy, may be due either to negative running, or may haveanother more exotic origin. In this paper, we considereternal inflation in a scenario with nonzero running, andshow that a negative running of the scalar spectral indexon superhorizon scales serves to suppress eternal infla-tion. Assuming a constant running, we derive an upperbound α < − × − . (32)For running below this bound, the primordial power spec-trum is less than unity on all scales larger than the cur-rent horizon, and eternal inflation is prevented. In a morerealistic case where higher-order terms such as running-of-running become significant, it is still the case that, aslong as the curvature perturbation remains smaller thanorder unity, eternal inflation does not occur. In single-field inflationary models, negative running eventually re-sults in a breakdown of slow roll and therefore a finiteduration for inflation. We show that negative running isconsistent with as many as 10 e-folds of inflation, with- out the onset of eternal inflation, in contrast to the caseof no running, for which eternal inflation will occur givenaround 1000 e-folds of inflation.Can we really know what occurs very early in the in-flationary epoch? Eternal inflation requires a curvatureperturbation spectrum of at least order unity to occur, P ( k ) ≥ i.e. around 60 e-folds beforethe end of inflation) cannot give rise to eternal inflation,since CMB normalization requires P ( k ) ∼ − . Eter-nal inflation occurs on the portion of the potential wherethe inflaton field rolls prior to producing these perturba-tions, which corresponds to length scales larger than ourcurrent horizon size. For example, in an m φ potential,with m ∼ GeV, eternal inflation only takes placehigh up in the potential, at φ > M P . Since we do nothave observational access to superhorizon length scales,and therefore the physics of very early stages of inflation,any conclusion we might reach contains an inherent ele-ment of speculation: It may be that such high regions ofthe potential are never probed, for example in the caseof non-negligible spatial curvature [75]. In this paper, wehave not proven that eternal inflation does not occur. Wehave argued that it is not inevitable, even in single-fieldinflation, and current data in fact hint that we may bein a situation where eternal inflation is suppressed, evenon far super-horizon scales. Acknowledgments
KF acknowledges the support of the DOE under grantDOE-FG02-95ER40899 and the Michigan Center forTheoretical Physics at the University of Michigan. WHKis supported by the National Science Foundation undergrant NSF-PHY-1066278. We thank Alejandro Lopez forhelpful conversations. This work was performed in partat the University at Buffalo Center for ComputationalResearch. [1] A. H. Guth, Phys.Rev.
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