Multi-field inflation with random potentials: field dimension, feature scale and non-Gaussianity
PPrepared for submission to JCAP
Multi-field inflation with randompotentials: field dimension, featurescale and non-Gaussianity
Jonathan Frazer and Andrew R. Liddle
Astronomy Centre, University of Sussex, Brighton BN1 9QH, United KingdomE-mail: [email protected], [email protected]
Abstract.
We explore the super-horizon evolution of the two-point and three-point cor-relation functions of the primordial density perturbation in randomly-generated multi-fieldpotentials. We use the Transport method to evolve perturbations and give full evolutionaryhistories for observables. Identifying the separate universe assumption as being analogousto a geometrical description of light rays, we give an expression for the width of the bundle,thereby allowing us to monitor evolution towards the adiabatic limit, as well as providing auseful means of understanding the behaviour in f NL . Finally, viewing our random potentialas a toy model of inflation in the string landscape, we build distributions for observables byevolving trajectories for a large number of realisations of the potential and comment on theprospects for testing such models. We find the distributions for observables to be insensitiveto the number of fields over the range 2 to 6, but that these distributions are highly sensitiveto the scale of features in the potential. Most sensitive to the scale of features is the spectralindex, with more than an order of magnitude increase in the dispersion of predictions over therange of feature scales investigated. Least sensitive was the non-Gaussianity parameter f NL ,which was consistently small; we found no examples of realisations whose non-Gaussianity iscapable of being observed by any planned experiment. Keywords: inflation, non-Gaussianity, string theory and cosmology a r X i v : . [ a s t r o - ph . C O ] N ov ontents ζ N P ζζ , f NL and the adiabatic limit 124.2.1 P ζζ and Θ 124.2.2 f NL is always small 144.2.3 Trends in f NL evolution 164.3 Interlude: The Lyth bound 164.4 Distribution of observables: n and r With the prospect of improved data from the Planck mission fast approaching, there has beena lot of interest in finding inflationary models exhibiting specific observable footprints. Largenon-Gaussianity signals peaking at various shapes is one example, another being features inthe primordial power spectrum. While this is a crucial step towards understanding whatobservables are specific to a particular model, often the set-up can be somewhat contrivedand to gain understanding as to whether such behaviour is a general feature of the model,one may need to invoke Monte Carlo techniques. In such a situation it may be helpful toemploy some element of ‘randomness’ at the level of the construction of the model.For example, a popular model of inflation coming from string theory is Dirac–Born–Infeld inflation. The DBI Lagrangian is L = − T ( φ ) (cid:115) − XT ( φ ) + T ( φ ) − V ( φ ) , (1.1)– 1 –here X = − g µν ∂ µ φ∂ ν φ and T ( φ ) is the brane tension. If the D3-brane velocity approachesits limiting speed 1 − XT ( φ ) → , (1.2)then a period of inflation can occur. This model has interesting observational consequencesas the sound speed can become small and hence, since f (eq)NL ∼ /c , lead to large equilateral f (eq)NL . However the above Lagrangian also admits inflation by other means. In Ref. [1] a rathersophisticated model of brane inflation was investigated, where to simulate the effect of thebulk in different compactifications, random coefficients were used. In this set-up, conditionsfor DBI inflation were never encountered; instead inflection-point slow-roll inflation was vastlymore common. We therefore see that while the DBI effect certainly gives an interestingobservational footprint, there is no reason to believe this is a generic feature of brane modelsof inflation. On a more ambitious note, string theory seems to predict the existence of a landscape [2, 3],where, in the low-energy approximation, different regions may be characterised by the val-ues of a large number of scalar fields. The consequence of this is that we have some verycomplicated potential V ( φ , ..., φ d ) with a large number of minima each corresponding to adifferent metastable vacuum energy. This implies that instead of trying to predict the valuesof observables, we should be trying to predict probability distributions for them. Indeed, aswe will now discuss, this is the case not just for string theory but for any model with multiplelight fields.Most work on the consequences of a landscape has focussed on the measure problem(see Ref. [4] for a recent overview) but if the observational consequences of such a modelare ever to be understood, then there are other challenges to contend with. In order for alandscape model (any model where the scalar potential has more than one minimum, or forthe purposes of this discussion, even just one minimum but multiple fields) to be predictive,three questions need to be addressed:1. What are the statistical properties of the landscape What are the selection effects from cosmological dynamics What are the anthropic selection effects
The measure problem relates to the question of handling the numerous infinities whichturn up. Taking the example of slow-roll inflation, any model of multi-field inflation suffersfrom an uncountably infinite set of choices for initial conditions. In general one needs toassume that, one way or another, at some point our region of spacetime experienced fieldvalues displaced from our local minimum. This corresponds to a single realisation of initialconditions (plus quantum scatter) that gave an anthropically suitable inflationary trajectory,which subsequently found its way to our local minimum. In order to make predictions we As pointed out in Ref. [1], this result is not conclusive since, rather importantly, their investigation didnot go all the way to the tip of the throat. Nevertheless we feel this example illustrates the point in hand. Most discussion in this area focuses on the scenario of inflation coming from tunnelling between metastablevacua but as we discuss here, the problem is much more general than that, affecting even the most pedestrianof inflationary set-ups. – 2 –eed to ask what proportion of the whole universe finds itself in this situation, i.e. whatproportion of an infinite space finds itself in one of an infinite set of initial conditions. Theratio is ill-defined without a measure.However overcoming this formidable task is not the end of it. Even with a solution tothis measure problem we are still left with a considerable challenge. A solution to this issueis likely not to give us a specific set of initial conditions for a given model but a probabilitydistribution for them. If all we can hope for is a statistical description of initial conditions,then in turn we only have a statistical description of inflationary trajectories and so, ratherthan calculating single values for observables, we should be calculating their distributions!The shape of these distributions will in part be determined by the model. This last pointcan, at least in some respects, be studied in its own right without a detailed knowledge of thestring landscape or the measure problem. In this paper we take inspiration from the stringlandscape and study characteristics of these distributions in the context of a potential withmultiple fields, containing a large number of vacua.An early study of the possible consequences of this landscape picture for slow-rollinflation was carried out by Tegmark [5], who generated a large number of random one-dimensional potentials and explored the inflationary outcomes. In Ref. [6] we extended thisto two fields to investigate the effect of entropy modes on super-horizon evolution. As al-ready mentioned, in Ref. [1] a similar analysis was done for a six-field brane inflation modelwith random terms arising in the contribution coming from the bulk, where although thesuper-horizon effects were not analysed, both reassuringly and rather excitingly, qualita-tively similar emergent behaviour was identified to that found in Ref. [6]. In this paper wefurther extend our work in Ref. [6] to a larger number of fields and a broader range of po-tentials, as well as obtaining results for the non-Gaussianity f NL . The aim is to gain insightinto the origin and limits of emergent behaviour. We construct our potential following an approach similar to Refs. [5, 6]. We use a randomfunction of the form V ( φ ) = m (cid:88) 10 10 20 Φ (cid:45) (cid:45) Φ (cid:45) (cid:45) 10 10 20 Φ (cid:45) (cid:45) Φ Figure 1 . Example trajectories for our potential (blue) and Fourier series (red) for two-field modelwith k max = 3. The potentials we simulate are periodic with periodicity scale 2 πm h , and we can onlyexpect reasonable results if the field trajectory spans a distance in field space less than theperiodicity of the random function. This turns out to always be the case.Note that summing the potential this way means we are not using the most generalFourier series. As shown in Fig. 1, by restricting the summations over each k i to non-negative values we sacrifice statistical d -spherical symmetry but in doing so we are able tobuild an observationally indistinguishable potential out of a fraction of the number of terms(see Table 1 for the number of terms in the series for various d and k max values). As discussed, any model of inflation where the potential has multiple minima predicts a prob-ability distribution for the cosmological parameters. We wish to compute this distributionfor various potentials of the above form. To do this we perform the same experiment as thatperformed in Refs. [5, 6]:1. Generate a random potential V ( φ ) and start at φ = (0 , V (0 , < N < 60 we rejectas insufficient inflation occurred, otherwise calculate observables.5. Repeat steps 1-4 many times to obtain a statistical sample.6. (Change some assumptions and do it all again.)– 4 –ur Potential Fourier SeriesFields d k max Table 1 . Summary of how the number of terms in the potential changes with the truncation k max and number of fields d for Fourier series potential and our reduced version. Note that due to our potential being statistically invariant under translation, generatingmultiple realisations of the potential and starting at the origin is equivalent to taking asingle realisation and scanning over initial conditions.In Ref. [6], taking the final minimum as the ultimate vacuum energy, to give an ap-proximately anthropically suitable solution [7] we had an additional cut stipulating the finalvacuum energy must be positive to avoid subsequent collapse. This, in conjunction with therejection of eternally inflating vacua, was found to be an extremely severe cut, in some casesreducing the proportion of otherwise viable solutions from 0.06 to more like 2 × − . In thispaper we abandon this cut, to enable us to explore more featured potentials which wouldnot otherwise be computationally accessible. We found this to be of little consequence forobservables. An explanation for this is that the two models may differ only in the natureof the post-inflationary evolution of the trajectory, which has no effect on the evolution ofobservable quantities.The other consideration regarding the experimental set-up is at what value to set thevertical and horizontal mass scales m v and m h . The vertical mass has little dynamical impactand only affects the amplitude of the observed power spectrum by a factor and not otherobservables. For this reason, rather than fixing m v we adjust it on a case-by-case basis suchthat the amplitude at horizon exit is P ∗ ζ = 2 × − .The horizontal mass m h is more interesting. As previously discussed, motivated by theaim of minimising the number of terms in the potential for a given dynamical behaviour, therandom coefficients are chosen in such a way as to make the potential essentially insensitiveto truncation. This set-up means that m h is our key parameter in adjusting how featured thepotential is. As the examples in Fig. 2 show, adjusting the scale of features affects the lengthscale ∆ φ of the inflationary distance in field space. We will discuss motivation from theoryfor this length scale next, but it is important we understand its implications for predictabilityand thus we shall be showing results for a range of m h values. Generally one expects that the inflationary potential can be well described by an effective the-ory containing non-renormalizable contributions coming from integrating out massive fields. For instance, for a single-field model one can write V ( φ ) = V + 12 m φ + M φ + 14 λφ + ∞ (cid:88) d =5 λ d M (cid:18) φM Pl (cid:19) d , (2.3)where the terms in the summation are non-renormalizable. One expects the masses in thesummation to be at or even well below the Planck mass as, in analogy to the argument Discussion along these lines can be found in Ref. [8] and of course Ref. [9] but we would particularly liketo thank Liam McAllister and Sam Rogerson for very helpful clarifications and additional comments on thismatter. – 5 – igure 2 . Example of two-field potentials with m h = 15 . M Pl and m h = 2 . M Pl respectively. from W – W scattering for a Higgs around 1 TeV, there needs to be something to unitarizegraviton–graviton scattering. There is no good reason to assume the inflaton does not coupleto these extra degrees of freedom. To do so is to make a strong assumption about quantumgravity which is hard to justify, and thus we expect λ d ∼ 1. If we categorise inflation modelsas large field, | ∆ φ | (cid:29) M pl , medium field | ∆ φ | ∼ M pl and small field | ∆ φ | (cid:28) M pl , thenthis sort of reasoning indicates small and perhaps medium field models should be consideredmore realistic as terms in the summation are suppressed, while to have a large-field model,one needs to justify additional symmetries to protect the flatness of the potential againstthe otherwise increasingly large series contributions. Crudely speaking we can think of ourchoice of m h as corresponding to a decision on what energy scales we are integrating out.Finally, we would like to consider the number of fields to be included in the model.Historically a lot of focus has been given to single-field models simply because they are themost basic inflationary set-up, but this is not what is best motivated from the field theoryperspective. As already mentioned, a single-field model occurs when one degree of freedomis much lighter than all the others. This means one can integrate out the other degrees offreedom provided they are sufficiently massive, but there is no good reason to believe thisis necessarily the case. For example, in string theory the contributing massive fields includestabilised moduli. Work on flux compactifications is still very much in development buttypically masses correspond to around the Hubble scale. This strongly motivates modelswith tens if not hundreds of active fields [10].It therefore seems quite reasonable to model the final inflationary phase in a landscapeas a truncated d -field Fourier series with random coefficients, provided we are dealing withsmall- to medium-field models. However, for computational reasons we are forced to workwith something less realistic. Ideally we would work with more fields, and push to smallerfield excursions than we will be working with. For the purposes of our investigation we willat times be working with inflationary trajectories that not particularly well motivated asgenuine models of inflation, yet we still find them to be quite informative when it comes tounderstanding inflationary dynamics. – 6 – The Transport equations In this paper, we improve on our previous work [6] by calculating the perturbations using theTransport method of Mulryne et al. [11, 12]. This gives improved computational efficiencyfor the power spectrum, while still including all isocurvature effects, and additionally allowsus to compute the non-Gaussianity parameter f NL . We compared results from this methodwith the geometric approach (see Ref. [13] for early work in this area; see Ref. [14] for somemore recent work) used in our previous paper [6], and they were found to agree for all modelstested, as well as giving the same distributions of observables when tested on our landscapemodel.With regard to calculating f NL , compared to other methods (for instance Ref. [15, 16]),the Transport approach has the benefit of being computationally more efficient, as wellproviding a new means of understanding contributions to f NL by having explicit sourceterms. Equivalent to all other methods in the literature (including cosmological perturbationtheory), it is simply an implementation of the separate universe assumption, but instead ofevolving many perturbed trajectories, as is done in the popular δN approach [15–17], oneevolves probability distributions.What follows is largely a summary of the work done in Refs. [11, 12]. We focus onexplicitly showing how the Transport formalism is implemented for a general d -field modelof inflation and refer the reader to Refs. [11, 12] for the details. ζ In calculating the statistical properties of the curvature perturbation we invoke the separateuniverse assumption and consider a collection of space-time volumes whose mutual scatterwill ultimately determine the microwave background anisotropy on a given scale. Each space-time volume follows a slightly different trajectory in field space, whose position at a giventime we label φ ∗ , the scatter of which is determined by the vacuum fluctuations at horizonexit. Here and in what follows the superscript “ ∗ ” indicates that the quantity is evaluated ona spatially-flat hypersurface. If we know the distribution P ( φ ∗ ) then, among other things, wecan study the statistical properties of the deviation of these trajectories from their expectationvalue Φ, δφ ∗ i = φ ∗ i − Φ ∗ i , where i indexes the components of the trajectory φ , namely thespecies of light scalar fields. The two-point correlations among the δφ i are expressed by thecovariance matrix Σ( t ), where Σ ij ≡ (cid:104) δφ i δφ j (cid:105) (3.1)and the third moment is given by α ijk ≡ (cid:104) δφ i δφ j δφ k (cid:105) (3.2)The covariance matrix, third moment and centroid Φ are all functions of time, but in ournotation we will be suppressing the explicit time dependence.A consequence of the separate universe assumption [17] is that the curvature pertur-bation ζ evaluated at some time t = t c is equivalent on large scales to the perturbation ofthe number of e-foldings N ( t c , t ∗ , x ) from an initial flat hypersurface at t = t ∗ , to a finaluniform-density hypersurface at t = t c , ζ ( t c , x ) (cid:39) δN ( t c , t ∗ , x ) ≡ N ( t c , t ∗ , x ) − N ( t c , t ∗ ) (3.3)– 7 –here N ( t c , t ∗ ) ≡ (cid:90) c ∗ Hdt. (3.4)Expanding δN in terms of the initial field perturbations to second order, one obtains ζ ( t c , x ) = δN ( t c , t ∗ , x ) = N ,i δφ ∗ i + 12 N ,ij ( δφ i δφ j − (cid:104) δφ i δφ j (cid:105) ) , (3.5)where repeated indices should be summed over, and N ,i , N ,ij represent first and secondderivatives of the number of e-folds with respect to the fields φ ∗ i . We remind the reader thatit is necessary to subtract the correlation function in the second term. This is because onecan interpret the covariance matrix as the contribution from disconnected diagrams whichgives the vacuum energy. In Fourier space one only considers connected diagrams from theoutset and thus the subtraction is already implicitly taken care of.Combining Eq. (3.1) and Eq. (3.2) with Eq. (3.5) we get expressions for the two- andthree-point functions in terms of the moments of P ( φ ∗ ). The two-point function is (cid:104) ζζ (cid:105) = N ,i N ,j Σ ij . (3.6)It is useful to decompose the three-point function as (cid:104) ζζζ (cid:105) = (cid:104) ζζζ (cid:105) + (cid:104) ζζζ (cid:105) , (3.7)where (cid:104) ζζζ (cid:105) = N ,i N ,j N ,k α ijk , (3.8)and (cid:104) ζζζ (cid:105) = 32 N ,i N ,j N ,km [Σ ik Σ jm + Σ im Σ jk ] . (3.9)Eq. (3.8) is the intrinsic non-linearity among the fields, while Eq. (3.9) encodes the non-Gaussianity resulting from the gauge transformation to ζ ; as one evolves from one flat hy-persurface to another, turns in the trajectory will contribute to the non-Gaussianity. This,as well as any non-Gaussianity present at horizon exit, is what is encapsulated in Eq. (3.8).However, this super-horizon evolution also causes the hypersurface of constant density tochange and so the gauge transformation from the flat hypersurface to the coinciding surfaceof constant density also contributes to the non-Gaussianity and this contribution is takeninto account in Eq. (3.9). N From Eq. (3.8) and Eq. (3.9), it is clear that in order to calculate moments of the powerspectrum we need a method for calculating derivatives of N . In general when using the δN technique it is difficult or impossible to find an analytic expression for the derivatives of N . Itis therefore necessary to run the background field equations many times from perturbativelydifferent initial conditions, stopping at some value for H which is the same for all the runs.One then calculates the derivatives of N with respect to the initial conditions. In using theTransport equations, however, this process is replaced by solving a set of coupled ordinarydifferential equations. Instead of taking the surfaces “ ∗ ” and “ c ” to be at horizon crossing and– 8 –ime of evaluation respectively, instead the surfaces are taken to be infinitesimally separatedand the transport equations evolve the field values at horizon crossing forward to the timeof evaluation. The upshot of this is two-fold. As we will see, the use of ordinary differentialequations to evolve the moments of the field perturbations allows us to see the source ofsuper-horizon evolution and hence the various contributions to f NL . The second and moreimmediate benefit to our current discussion is that we can find a general expression for thederivatives of N . To leading order in slow-roll, for a given species “ i ”, the number of e-folds N between the flat hypersurface and a comoving hypersurface is given by N ( t c , t ∗ ) ≡ − (cid:90) φ c φ ∗ VV ,i dφ i , no sum on i . (3.10)and so if the two surfaces are infinitesimally separated, then we can write dN = (cid:20)(cid:18) VV ,i (cid:19) ∗ − (cid:18) VV ,j (cid:19) c ∂φ cj ∂φ ∗ i (cid:21) dφ ∗ i . (3.11)To handle ∂φ cj /∂φ ∗ i the method used in Refs. [16, 18] for sum-separable potentials is also nowapplicable and we introduce the quantity C i ≡ − (cid:90) dφ i V ,i + (cid:90) dφ i +1 V ,i +1 . (3.12)This enables us to write dφ ci = ∂φ ci ∂C j ∂C j ∂φ ∗ k dφ ∗ k (3.13)which after some algebra gives the expression ∂φ ci ∂φ ∗ j = − (cid:18) VV ,j (cid:19) ∗ (cid:18) V ,i V (cid:19) c (cid:32) V c,j V c,k V c,k − δ ij (cid:33) (3.14)Hence we find N ,i = (cid:18) VV ,i (cid:19) ∗ (cid:32) V ,i V ,k V ,k (cid:33) c , no sum on i (3.15)and N ,ij = V ,i V ,j V ,k V ,k + V V ,ij V ,k V ,k − V V ,ik V ,k V ,j ( V ,k V ,k ) − V V ,jk V ,k V ,i ( V ,k V ,k ) + 2 V V ,i V ,k V ,kl V ,l V ,j ( V ,k V ,k ) , (3.16)where in Eq. (3.16) the limit c → ∗ has been taken. Time of evaluation is often taken to be the end of inflation but with regard to calculating observables,any time after isocurvature modes have decayed away will give the same result. A problem arises whenisocurvature modes are still present at the end of inflation. In this case the power spectrum will continueto evolve and without a model of reheating this renders the model non-predictive. We will return to this ingreater detail later on. – 9 – .3 Transporting the moments Finally, we need a method for evolving the moments of the scalar perturbations. The proba-bility distribution P ( φ ∗ ) is conserved and so, as described by the standard continuity equa-tion, the rate of change of P is given by the divergence of the current, ∂P∂N + ∂ ( u i P ) ∂φ i = 0 , (3.17)where u i ≡ φ (cid:48) i is the field velocity. The key achievement of Ref. [11] was to develop a methodfor extracting the evolution equations of the moments of P from the continuity equation.In Ref. [12] an alternative method was introduced, generalising to any number of fields onarbitrary slicing. We do not go into the techniques here; instead we just quote the resultingevolution equations for the centroid, variance and skew which collectively we refer to as theTransport equations, Φ (cid:48) i = φ (cid:48) i + 12 u i,mn Σ mn + ... , (3.18)Σ (cid:48) ij = u i,m Σ mj + u j,m Σ mi + 12 u i,mn α jmn + 12 u j,mn α imn + ... , (3.19) α (cid:48) ijk = u i,m α mjk + u i,mn Σ jm Σ kn + (cyclic i → j → k ) + · · · . (3.20)The equation for the centroid Eq. (3.18) says that the mean field value evolves as the velocityof the fields but can be affected by evolution of the wings of the distribution. The evolu-tion equations for the variance and skew, as one might guess from the continuity equationEq. (3.17), give evolution as the divergence of the field velocity but now also with sourceterms coming from the other moments. As will be discussed in more detail in due course, an important consideration in our analysiswill be whether or not evolution of observables is still taking place at the time of evaluation.Evolution stops when the trajectory becomes effectively single field [19]. This is to say thetrajectory has reduced to a caustic [20], so for this reason we would like a description for theevolution of the cross-section of the perturbed trajectories. Such a description has recentlybeen developed in Ref. [21], to which we refer the reader for more detailed discussion. Forsimplicity we only describe the broad concept here and quote results that will be needed infuture discussion.Cross-sections within the bundle are focused, sheared and rotated by the flow. Thesedistortions can be characterised by the evolution of connecting vectors describing the dis-placement between nearby trajectories in the bundle. If δx i is an infinitesimal connectingvector, then assuming u i,j is sufficiently smooth, δx i is transported as˙ δx i = δx j ∂u i ∂φ j (3.21)It follows that changes in the cross-section of the bundle can be determined in terms of theexpansion tensor u i,j . We can decompose this in terms of a dilation θ = tr u i,j , a tracelesssymmetric shear σ ij , and a traceless antisymmetric twist ω ij , u i,j ≡ θd δ ij + σ ij + ω ij (3.22)– 10 –ilation describes a rigid rescaling of δx i by 1 + θ , representing a global tendency of thetrajectories to focus or defocus. The shear encapsulates the tendency for some trajectoriesto flow faster than others while conserving the cross-sectional area of the bundle. The twistrepresents a rotation of the bundle with preserved volume, such as the tendency of trajectoriesto braid. The dilation, shear and twist act as sources for one another and so one expects abundle will typically exhibit all of these behaviours at some point.We refer the reader to Ref. [21] for a more formal description of this formalism and itsapplications, but for the purposes of this paper all we need is the result that the focussingof the bundle is given by Θ( H, H ) = exp (cid:20) d (cid:90) HH θ ( h ) dh (cid:21) . (3.23) Having set up our models and the machinery necessary to compute the observables, we nowproceed to our results. The principal variables of interest to vary are the number of fields d and the horizontal mass scale m h . Large values of the latter correspond to relatively smoothpotentials, and small values to heavily featured potentials. We refer to individual realizationsgiving sufficient inflation as ‘verses’.We discuss our results in the following sequence:1. Dynamical properties of trajectories.2. Perturbation evolution along individual trajectories.3. Distribution of observables over ensembles of trajectories. d dependence Fig. 3 summarises the qualitative behaviour found during ourexploration of the properties of multi-field trajectories. We see that ∆ φ , the length of thetrajectory in field space, and B φ , the percentage increase in ∆ φ due to turns in the trajectory,defined as B φ ≡ 100 ∆ φ − (cid:112) φ end · φ end − φ ∗ · φ ∗ ∆ φ (4.1)show only a mild sensitivity to the number of fields. In fact, for all observables and infla-tionary parameters we looked at, the sensitivity to changing the number of fields was smallover the range d = 2 to d = 6 compared to the spread of results. This was true even forthe relatively predictive large-field case of m h = 15 M Pl . Least sensitive of all, we foundno change whatsoever in the distribution of slow-roll parameters at horizon crossing with (cid:15) ∗ = 0 . ± . 002 and η ∗ = 0 . ± . e-fold distributions One instance where we did find sensitivity to d was in the e-folddistributions, where we saw a decrease in the proportion of trajectories with more than 60e-folds from 0.08 for two fields down to 0.01 for six fields. Fig. 3 hints at a tendency fortrajectories to become shorter and more curved as the number of fields increases. So giventhat the trajectory will seek the route of steepest descent, it appears that for our model,increasing the number of fields increases the chance of the trajectory encountering a slopesufficiently steep to kill inflation. – 11 – d (cid:68)Φ d B Φ (cid:72) (cid:37) (cid:76) Figure 3 . How the mean ∆ φ and percentage increase in ∆ φ due to bending of the trajectory changeswith the number of fields, for m h = 15 M Pl . The bars show how the standard deviation of thesequantities changes over this range. In Refs. [1, 3] slightly different parameterisations of the inflationary region of the po-tential were used to show the probability of obtaining a given number of e-folds of inflationwas P ( N ) ∝ N α , (4.2)where α = 4 and α = 3 were found respectively. Bearing Eq. (2.3) in mind, it seemsreasonable that Eq. (4.2) might apply more generally and indeed we find it to be a good fitto our e-fold distributions, with α increasing as the potential becomes more featured from α = 2 through to α = 5 over the range of m h we investigated. To illustrate the implication ofthis, consider the popular idea that inflation was preceded by a tunnelling event. Using the(perhaps somewhat arbitrary) values of Ref. [3], we place an anthropic lower bound on thenumber of e-folds at 59.5 coming from structure formation and an observational lower boundat N = 62 on the curvature from tunnelling. Then for α = 2, the probability of the modelachieving sufficient inflation to be in agreement with observation is roughly 92%, while for α = 5 it is more like 81%. However, remember we are working in the range of ∆ φ which is notbest motivated theoretically. While not accessible with the techniques used here, our resultslead us to expect that for small-field models, α should be larger. This has the potential tocause tension with observation, as by α = 18 the chance of finding ourselves in the observeduniverse falls to 49%, i.e a typical observer would expect to see evidence of curvature. P ζζ , f NL and the adiabatic limit Having seen that our model is insensitive to the number of light fields, for simplicity we onlygive results for two-field potentials in the remaining sections of this paper, focussing mainlyon the dependence on the feature scale of the potential. But we would like to emphasise thatthe results hold more generally. P ζζ and ΘThe key difference between single-field and multi-field inflation is that the latter admitsevolution of the power spectrum on super-horizon scales. This means that in order to make aprediction from multi-field inflation one needs to know the full evolutionary history up untilthe model becomes effectively single field, i.e. the adiabatic limit is reached [19, 20]. Oncethis happens the power spectrum stops evolving and one can evaluate observable quantities– 12 – P ΖΖ m h (cid:61) 100 200 300 400 N0.000050.000100.000150.000200.000250.00030 P ΖΖ m h (cid:61) P ΖΖ m h (cid:61) P ΖΖ m h (cid:61) Figure 4 . Example plots showing the super-horizon evolution of the power spectrum for modesexiting 55 e-folds before the end of inflation. m h = 3 M Pl corresponds to a highly-featured potential,and in all such cases evolution stops long before the end of inflation. m h = 15 M Pl is a comparativelysmooth landscape, and then a significant proportion of verses are still evolving at the end of inflation. at a subsequent time of one’s pleasing. The problem is that there is no guarantee that such anadiabatic limit will be reached before the end of inflation, and if this is not the case makinga prediction requires knowledge of reheating and so forth.As can be seen in Figs. 4, 5 and 6, the ability to reach the adiabatic limit, where theperturbation on a given scale becomes constant, is strongly dependent on how featured thelandscape is. For the more featured landscapes such as m h = 3 M Pl , we found the adiabaticlimit was reached in all cases (even the trajectories disappearing off the top of the plots),while for the smoother landscapes like m h = 15 M Pl the proportion of trajectories achievingthis clearly drops significantly.There is a very intuitive reason for why this should be the case. Rewriting Eq. (3.23)in terms of e-folds N we findΘ( N, N ∗ ) = exp (cid:20) d (cid:90) NN ∗ (3 (cid:15) − η + tr M ij ) dN (cid:21) (4.3)where M ij is the Hessian of ln V and ¯ η is the generalised slow-roll parameter¯ η ≡ V ,i V ,j V ,ij V V ,k V ,k (4.4)We therefore see that in a valley, strong focussing will occur, while on a ridge or a hilltop thebundle will dilate. With this picture it is quite easy to see why we should expect evolutionas seen in Fig. 4. The treacherous landscape of m h = 3 M Pl typically gives exactly the– 13 – (cid:45) (cid:45) (cid:45) f NL m h (cid:61) f NL m h (cid:61) Figure 5 . Example plots showing the super-horizon evolution of f NL for modes exiting 55 e-foldsbefore the end of inflation. Again we see that for the more featured landscape evolution stops earlyon, while for the smoother example, evolution often continues to the end of inflation. Figure 6 . Θ at the end of inflation for m h = 3 M Pl (blue), m h = 9 M Pl (yellow) and m h = 15 M Pl (red). All trajectories essentially reach a caustic in the most featured example, less for m h = 9 M Pl and least for the smoothest landscape m h = 15 M Pl . conditions required for a very strong focusing, while in contrast the comparatively mild,undulating meadows of m h = 15 M Pl give very little incentive for trajectories to focus to acaustic. f NL is always small Much of this kind of discussion carries over to understanding the results of Fig. 5. Firstand foremost it should be noted there was not a single example of a trajectory that gavesufficient non-Gaussianity to be detected by any future planned experiment. Methods to getaround this disappointingly generic feature of multi-field inflation were recently addressed inRef. [20] and the special case of sum-separable potentials was also discussed in Ref. [22]. Inthe case of sum-separable hilltop potentials which reach an adiabatic limit during inflation,what is known as the horizon-crossing approximation [23] gives a good estimate of the final– 14 – (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) f NL , f NL1, f NL2 20 30 40 50 N (cid:45) (cid:45) (cid:45) f NL , f NL1, f NL2 20 30 40 50 N51015 (cid:227) (cid:217) Θ dH 20 30 40 50 N12345 (cid:227) (cid:217) Θ dH Figure 7 . Plots of Verse 113147 (left) and Verse 253911 (right) showing very distinctive f NL evolu-tion. In the plots on the top row, f NL is blue, the intrinsic component, f NL1 , is red and the gaugecontribution, f NL2 , is yellow. value for f NL . f NL ≈ − V (cid:48)(cid:48) φ V φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ , (4.5)where in this instance φ represents one or at most a few fields where N i is large. From thiswe see that provided there are enough fields to keep η small, with the right initial conditions,a ridge can give rise to a large f NL .What we find in our analysis is that the problem of obtaining a large f NL is madeparticularly acute by the need to obtain sufficient inflation. When we have a very smoothlandscape, sufficient inflation is easily achieved but the lack of features means there is nothingto give rise to a large f NL . On the other hand, when the potential is very featured, it isdifficult to start close enough to a ridge to get a large f NL without falling off it, therebykilling inflation. In some models, such as axion N-flation [24], a sufficiently large number offields can make it possible to overcome this problem [25], as the large damping term makes iteasier to be close to a ridge without falling off too soon. For our model, while we do not havethe computational power to explore this possibility, with enough fields we would expect tosee some examples with a large f NL , but they would constitute only a very small proportion.This is because if many fields have a large N ,i , their contributions to f NL will, through amanifestation of the central limit theorem, cause f NL to be vanishing in the limit of manycontributing fields. Thus on average we would always expect f NL to be small.– 15 – .2.3 Trends in f NL evolution While we found no examples of large f NL we did find a very diverse range of behaviour.This diversity is indicative of why detection would be such a powerful constraint on models.That said we did find some common trends. Fig. 7 shows two examples of evolution of f NL together with the corresponding evolution in the width of the bundle. We chose these twoexamples, Verse 113147 and Verse 253911, in particular as each shows characteristics thatwere common to most trajectories, but also demonstrate that counter examples were found. The gauge contribution determines the peak As particularly well demonstrated byVerse 113147, in all examples we found f NL2 determined any peaks in f NL . Typically it wasthe case that the intrinsic non-Gaussianity played a highly subdominant role in the feature,but we did find the exception that is Verse 253911 which clearly received an importantcontribution from the intrinsic part. In Ref. [11] it was noted that for the double quadraticpotential and quadratic exponential potential this behaviour was present. Here we show thischaracteristic applies much more generally. f NL grows when the bundle dilates Features in f NL occur whenever Θ grows. This seemsvery reasonable since at this point the perturbed trajectories will be exploring different partsof the potential. Typically the peak in f NL occurred very close to the time of the peak in Θbut again, Verse 253911 shows this need not be the case precisely. We see features in f NL2 are intimately related to features in θ . We attribute this to common terms involving V ,ij . Asymptotic behaviour of f NL is not straightforward A result that continues to eludeus is a simple way of understanding what the final value of f NL in the adiabatic limit willbe. For sum-separable hilltop potentials, the horizon-crossing approximation works well,but for more general potentials there is no equivalent. As the examples in Fig. 7 show, theasymptotic value can be reached in dramatically different manners. Worse still is the factthat in the adiabatic limit the intrinsic and gauge transformations need not settle to constantvalues. This indicates that a different set of parameters should be considered if we are tomake progress with this question. Before moving on to discuss distributions of observables, we would like to take a brief momentto discuss the relation between field trajectories and the tensor-to-scalar ratio, as it will behelpful to bear in mind in the subsequent discussion.Taking N CMB to be the number of e-folds between when fluctuations on CMB scalesleft the horizon and the end of inflation, we can obtain a d -field version of the Lyth boundby writing N CMB = (cid:90) φ end φ CMB √ (cid:15) dφ (cid:107) , (4.6)where we are integrating along the field trajectory. If we assume (cid:15) is either constant orincreasing over this period, then we have2 (cid:15) < ∆ φN CMB . (4.7)For single field inflation r = 16 (cid:15) but when there are more fields the curvature perturbationevolves on super-horizon scales, suppressing r and so r < (cid:15) [6]. We therefore see that the– 16 –yth bound remains essentially the same for multi-field models as in the single-field case r < (cid:15) < . (cid:18) N CMB (cid:19) (cid:18) ∆ φM Pl (cid:19) (4.8)Planck hopes to measure the tensor-to-scalar ratio with an accuracy of a few hundredths,hence has discovery potential if it is of order 0.1 or so. Comparing the Lyth bound withthe discussion in section 2.3 we therefore see that a detection would exclude all small- andmedium-field models if only one field is admitted. However as previously discussed, multi-field models are strongly motivated by fundamental theory. If we consider the extreme caseof sum-separable potentials then the discussion of Section 2.3 requires each ∆ φ i (cid:28) M Pl butthere is no restriction on the number of fields contributing during inflation; this was forinstance the motivation of the N-flation proposal [24]. Therefore, if we are to stay in the fieldtheory favoured regime of small-field models, a detection of r would place a lower bound onthe number of fields! Rewriting Eq. (4.8) in a more suggestive form we have r < . d (cid:18) N CMB (cid:19) ∆ φ i ∆ φ i M . (4.9) n and r Fig. 8 shows our findings for n and r at the end of inflation, also summarised in Table 2. Asthe landscape becomes more featured the viable trajectories become shorter and increasinglybendy. The Lyth bound tells us the distance travelled in field space places an upper bound onthe tensor-to-scalar ratio. Furthermore, bends in the trajectory cause super-horizon evolutionof the curvature power spectrum, while the tensor power spectrum is conserved, and so asFig. 8 shows, we see an increasingly strong suppression in r as we move to lower m h .As summarised in Fig. 6, a less featured potential reduces the chance of trajectoriesreaching their adiabatic limit. It is noteworthy that, despite this, the plots of the n – r planefor m h = 15 M Pl and m h = 18 M Pl show remarkable consistency for n and r at the end ofinflation. This might lead one to think that the super-horizon evolution is having negligibleeffect, but if we take the example of m h = 15 M Pl , as Table 2 shows, the mean increase in thefield trajectory from turning is only 4%, yet if we assumed a single-field approximation wasvalid we would obtain n = 0 . ± . 01 which is significantly different from the actual resultof 0 . ± . 01. If nothing else, these results show one should be exceedingly careful whenmaking single-field approximations. Fig. 9 compares distributions for the spectral index withthose obtained using a single-field approximation.The central concern regarding the possible existence of a landscape is whether or notsuch a model can be tested. As we have mentioned, a key challenge is the measure problem,another being our very limited understanding of fundamental theory. Once these problems arebetter understood though, we will still be left with distributions for observable quantities.No matter how well developed our understanding, it seems reasonable to assume that atsome level our ability to make predictions will be fundamentally limited by the details of thetheory. Our toy landscape illustrates this in a very explicit way. In a sense m h gives a wayof quantifying the complexity of each landscape. For our model, we see the spread of resultsfor the spectral index dramatically increases as we move to more featured landscapes, whilefor the tensor-to-scalar ratio the spreading is considerably less dramatic due to a suppressioncoming from the inevitable decrease in the length of the field trajectory ∆ φ . As we have seen f NL by contrast remains consistently small.– 17 – .88 0.90 0.92 0.94 0.96 0.98 1.00 1.02n0.000.050.100.150.200.250.30r m h (cid:61) m h (cid:61) m h (cid:61) m h (cid:61) m h (cid:61) m h (cid:61) Figure 8 . n – r plots for a range of m h , beginning with the more featured potentials. The curve showsthe WMAP7+all 95% confidence limit [26]. m h ( M Pl ) ∆ φ B φ (%) n r , 95% conf. f NL . ± . ± 18 1 . ± . 15 0 . r < . − . ± . . ± . ± 18 0 . ± . 07 0 . r < . 172 0 . ± . . ± . ± 13 0 . ± . 03 0 . r < . 081 0 . ± . . ± . ± . ± . 02 0 . r < . 066 0 . ± . . ± . ± . ± . 01 0 . r < . 074 0 . ± . . ± . ± . ± . 02 0 . r < . 079 0 . ± . Table 2 . Table of the mean distance in field space travelled in the last 55 e-fold of inflation for agiven m h , B φ , the mean percentage increase coming from bends in the trajectory, and correspondingresults for observables. We explored inflationary dynamics in randomly-generated potentials as well as the con-sequences for super-horizon evolution of perturbations. We found this exploration to be– 18 – .6 0.8 1.0 1.2 1.4 1.620406080 m h (cid:61) m h (cid:61) Figure 9 . Example plots comparing spectral index as obtained using single-field approximation (red)and as calculated taking evolution up to the end of inflation into account (blue). interesting primarily on two fronts.First, by exploring a very large number of inflationary trajectories, we encountered awide range of super-horizon evolution behaviour for P ζζ and f NL . The benefit of this wasthat it became easy to see what characteristics are generic and which are not. We foundthat peaks in f NL tend to be determined by the gauge contribution but behaviour was rathermore broad in the adiabatic limit, showing few trends. Understanding of non-Gaussianityis still in rapid development and so exploration of this kind can be very helpful in gaininginsight in how to progress towards something more concrete such as Ref. [21].We also found that keeping track of the easy-to-compute bundle width was extremelyinformative. By following how Θ changes along the trajectory, we were generally able tounderstand what qualities of the potential gave rise to super-horizon evolution. In particular,we found that peaks in f NL occur during regions of the potential that give rise to a dilationof the bundle. However, a more quantitative description awaits future development. Weemphasise that in order to make predictions in any multi-field model, one needs to performan equivalent analysis to ensure no evolution is taking place at the time of evaluation. Wefound that as the mean length of the field trajectory in field space increased, the chances ofreaching an adiabatic limit drastically decreased, rendering the larger field models essentiallynon-predictive without a model of reheating.Second, we looked at how varying the scale of features and the number of light fieldsaffected the ensembles produced for a given parameter. We found that landscapes where themean length of the field trajectory was large typically gave results consistent with currentobservational data (despite not necessarily reaching their adiabatic limit). However for morefeatured landscapes where the mean field trajectory was smaller, the spread in the spectralindex increased significantly. The spread in the tensor-to-scalar ratio did not increase sodramatically. This can be understood in terms of the Lyth bound which places an upperbound on the tensor-to-scalar ratio according to the length of the field trajectory. 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