Slepian models for Gaussian random landscapes

JHEP05(2020)142 Published for SISSA by Springer Received:December 26, 2019 Revised:April 27, 2020 Accepted:May 4, 2020 Published:May 27, 2020 Slepian models for Gaussian random landscapes Jose J. Blanco-Pillado,a,b Kepa Sousacand Mikel A. Urkiolaa aDepartment of Theoretical Physics, University of the Basque Country, UPV/EHU, 48080, Bilbao, Spain bIKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain cInstitute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holesˇoviˇck´ach 2, Prague, Czech Republic E-mail: [email protected],[email protected], [email protected] Abstract: Phenomenologically interesting scalar potentials are highly atypical in generic random landscapes. We develop the mathematical techniques to generate constrained random potentials, i.e. Slepian models, which can globally represent low-probability realizations of the landscape. We give analytical as well as numerical methods to construct these Slepian models for constrained realizations of a full Gaussian random field around critical as well as inflection points. We use these techniques to numerically generate in an efficient way a large number of minima at arbitrary heights of the potential and calculate their non-perturbative decay rate. Furthermore, we also illustrate how to use these methods by obtaining statistical information about the distribution of observables in an inflationary inflection point constructed within these models. Keywords: Cosmology of Theories beyond the SM, Stochastic Processes, Superstring Vacua ArXiv ePrint: 1911.07618 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP05(2020)142 JHEP05(2020)142 Contents 1 Introduction 1 2 Preliminaries for Gaussian random fields 3 3 Slepian models for constrained Gaussian random fields 4 3.1 Slepian models for critical points 5 3.2 Slepian models for inflection points 7 3.3 2D numerical implementation 9 4 Tunneling in a Gaussian random landscape 9 4.1 Statistics of the instanton action 13 4.1.1 Dependence with the height 13 4.2 Approximations for the calculation of the action 14 4.2.1 Thin wall approximation 14 4.2.2 Straight-path approximation 15 4.3 The lowest action 16 4.3.1 Exit angle 16 4.3.2 Estimating the lowest action 16 5 Inflation in a Slepian random landscape 18 5.1 1D inflection point inflation 19 5.2 Numerical inflection points in a 2D landscape 20 5.3 Statistics of inflationary parameters 21 6 Summary and conclusions 24 A Construction of Slepian models 27 A.1 Introductory remarks and some properties of Gaussian random variables 27 A.2 Conditioned Gaussian random vectors 27 A.3 Gaussian random fields 29 A.4 Useful correlations 29 A.5 The Kac-Rice formula and conditioned Gaussian random fields 31 A.6 Conditioned Gaussian random field for a critical point 31 A.6.1 Analysis of a conditioned 2D Gaussian field 35 A.6.2 Distribution of heights and eigenvalues of the Hessian at a critical point 35 A.7 Conditioned Gaussian random field for an inflection point 36 A.7.1 Probability distribution for the inflection point parameters 38 B Numerical implementation and tests of the probability distributions 38 B.1 Generation of Gaussian random fields: Karhunen-Lo`eve expansion 38 B.2 Numerical evaluations of critical points 39 B.3 Numerical evaluations of inflection points 41 – i – JHEP05(2020)142 1 Introduction The low energy description of many higher dimensional theories involve a large number of fields (moduli fields) that need to be stabilized. This is normally achieved by the existence of a potential that fixes the values of these fields to a local minimum of that potential function. A typical example of this procedure can be found in String Theory compactification scenarios. In particular, models of flux compactification have been shown to lead to an enormous set of possible 4dpotentials that can have many local minima. The typical number of moduli fields in these cases is quite large, reaching often the order of a few hundred. This makes prohibitively difficult to study these potentials in detail and one is forced to look for simple models where the field space has been truncated to a small subset of fields. Alternatively, one can try to study these models by taking a more statistical approach, where the scalar potential is regarded as a random field whose sample space is the set of 4dlow-energy effective potentials. These ideas have been pursued in relation to the study of the stability of critical points in these potentials in [1–3], as well as the description of cosmological models for the early universe in [4–6]. In many of these studies one is interested in particular points of the landscape such as, for example, a minimum with some value of its cosmological constant, or an inflection point with a particular set of conditions in its derivatives necessary for it to sustain inflation. However, depending on the restrictions imposed, it may be very difficult to obtain an example of the potential with these characteristics by producing random realizations of the scalar potential. Indeed, metastable de Sitter vacua and inflationary points compatible with observations are very rare in generic landscapes, with probabilities scaling as P∼exp(−Np f), where Nfis the number of scalar fields in the theory, and p > 0 is a number of order one [7–12]. To obtain realizations with the desired properties, one can of course use a Taylor expansion around the point in question and take into account the probability distribution for its coefficients [13,14]. However this becomes quite complicated as one increases the number of fields and the field range that one is interested in.1 Moreover, with this type of procedures it is not possible to capture correctly the global properties of the scalar potential, which are essential to study quantum decay processes in the landscape. Here we present a different strategy to generate these potentials that locally will be constrained to have a particular form, but that globally will still represent a faithful realization of the random landscape, the so-called Slepian models [16]. Several different methods have been suggested as a way to represent these random potentials in the landscape. In this paper we will concentrate on potentials described by Gaussian Random Fields (GRFs). This is based on the assumption that the 4dpotential can be thought of as a sum of many different terms, of classical and quantum origin, coming from the compactification mechanism rendering the final result a Gaussian random field. This type of models have also been studied in connection to the distribution of vacua and its stability [9,17,18] as well as inflation [13–15,19,20] in the landscape. As an illustration of the mathematical techniques presented here for the construction of constrained GRFs we develop Slepian models that are locally described by critical points (maxima, minima and 1For another method of generating a specific class of constrained Gaussian random fields, see [15]. – 1 – JHEP05(2020)142 saddle points) as well as inflection points and use these realizations to extract important statistical information about them. In particular, we will first study the quantum mechanical stability of local minima in these landscapes. In order to do so, we will compute numerically the decay rate of these minima using the quantum tunneling techniques first described in a series of papers in [21,22]. The result of this quantum instability is the creation of a bubble instanton that interpolates between the false vacuum and the true vacuum states. Using these Euclidean methods one can evaluate the probability of this decay channel and therefore estimate the lifetime of any specific vacuum. The calculation of these tunneling events in a multidimensional potential is however notoriously difficult. Recently some work on this direction has been done in relation to the stability of vacua in models with large number of dimensions in field space. It has been argued that the probability of the decay depends exponentially on the number of fields although the particular scaling is still uncertain [23–26]. In this paper we will study these tunneling events in models of Gaussian random potentials. In particular we are interested in studying the dependence of the tunnelling rate with the height of the potential at the false vacuum. For large values of the cosmological constant this calculation would be impossible without constraining methods, since the number of these minima is negligible compared to the minima at lower values of the field. Our techniques allowed us to efficiently generate the same number of minima for different heights and have a good sample of cases from where we can extract statistical information. The obtained distribution for the instanton actions SE(which determines the decay rate Γ∼e−SE) is displayed in figures 5and 6, where we found that the average dependence of the decay rate on the false vacuum height Vfv is given by log10 SE U−1 0Λ4≈3.29 exp −0.18Vfv U0, where U0and Λ are the characteristic energy and length scale (in field space) of our potential respectively. The distribution for the Euclidean action becomes increasingly peaked around its mean, and thus more predictive, for larger values of Vfv. As we show in the main text, this enhancement of the predictability can be explained using Slepian models for very atypical extrema of the potential, such as high minima. Our second application involves the generation of inflection points. These are some of the most likely points in the landscape where cosmological inflation can happen. However this does not mean that an arbitrary inflection point would lead to inflation. Obtaining a successful inflationary period consistent with the current cosmological observations still requires some amount of fine tuning of the potential around the inflection point. Therefore, to characterise the distribution of observables for these inflationary models in the landscape one should again use some sort of constraining method, and look at a particular set of non-generic inflection points. In the present paper we will explore the dependence of the observable parameters of inflation to its initial conditions in the landscape. In particular we will take the initial conditions for the fields to be the ones determined by the exit point of an instanton describing the transition from a nearby parent false vacuum. Note that in order to perform this analysis, one requires not only the knowledge of the potential around the – 2 – JHEP05(2020)142 inflection point but also its relation to nearby minima. Hence our method, which accurately captures the global statistical properties of the potential, is particularly suitable to carry out this investigation. It is worth noting that, to the best of our knowledge, this is the first time that an Slepian model for inflection points is presented in the literature. The effect of the tunneling in the initial stages of inflation has also been discussed in [14,27–29]. The remaining of the paper is organized as follows. In section 2we introduce the notation that we will be using for describing our random potential function as a GRF. In section 3we will outline the method for generating constrained random potentials as Slepian models. In section 4, we implement these ideas for a 2dfield space landscape and generate a large set of random potentials with a minimum at a specific point in field space. This allows us to compute the tunneling paths from these minima and determine the statistics of the decay rate. In section 5, we condition the random potential to have an inflection point suitable for inflation, and study the effect of the initial conditions set by the tunneling process from a nearby minimum. We conclude in section 6with some comments on the results and some further ideas that can be implemented with these numerical techniques. Some of the mathematical details and numerical proofs have been left for the appendices. In the present work, unless otherwise stated, we will use reduced Planck units M−2 pl = 8πG/(~c) = 1. 2 Preliminaries for Gaussian random fields In this paper we will take our random potential, V(φ), to be a Gaussian random field defined over a N-dimensional field space, which we will parametrize with the vector φ= {φi}, with i= 1, . . . , N. Furthermore, we will consider the probability distribution for the random potential to be homogeneous and isotropic, so its covariance function will only depend on the distance between the points at which it is evaluated, in other words it is of the form hV(φ1)V(φ2)i=C(|φ1−φ2|).(2.1) We will additionally require the potential to have a null mean: hV(φ)i= 0 .(2.2) In the rest of the paper we will evaluate our expressions using the following simple covariance function: C(φ) = U2 0exp −φ2 2Λ2,(2.3) for the case of N= 2 field space dimensions. The parameter U0sets the energy scale of the potential while Λ represents the correlation length in field space. It is important to realize that the techniques used in this paper are generic and can be applied to other interesting situations like, for example, non-Gaussian covariance functions so in this sense these constructions are quite more generic than the ones presented in [15]. We have decided – 3 – JHEP05(2020)142 to use the simple Gaussian covariance function since it considerably simplifies some of the expressions in this paper.2 In the following we will be interested in the value of the field and its derivatives at a particular point in field space, which we can take to be φ= 0 without loss of generality, and we will refer to it as the center of field space. In order to simplify the notation we introduce the following definitions for the value of the potential and its derivatives: u=V(φ)|φ=0, ηi=∂V (φ) ∂φiφ=0 , ζij =∂2V(φ) ∂φi∂φjφ=0 , ρijk =∂3V(φ) ∂φi∂φj∂φkφ=0 . Furthermore, we will denote the eigenvalues of the Hessian matrix by λiwith i= 1,2 which will single out the directions 1,2 in our field space. Note that the derivatives of the scalar potential are also Gaussian random variables, and therefore any collection of the previous quantities forms a Gaussian random vector. In appendix A.4 we will give the expressions for the correlators between these different derivatives of the potential as a function of the derivatives of the covariance function C(φ). These correlations will play an important role in some parts of our discussions. 3 Slepian models for constrained Gaussian random fields A key point in our construction of the GRF rests on the fact that a conditioned GRF maintains its Gaussian nature. More specifically, homogeneous and isotropic processes (such as the GRFs we are dealing with) can be conditioned using the Kac-Rice formula [30] in order to obtain new mean and covariance functions which generate GRFs with the required constraints.3The models for stochastic processes dealing with conditional events and crossings where pioneered by David Slepian [16], and have thus been coined in the mathematical literature as Slepian models. We can describe these constrained processes in a generic form in the following way. For simplicity, let us consider first a Gaussian random p-dimensional vector, composed of jointly Gaussian variables, xT= (x1, . . . , xp), whose probability distribution function (PDF) is given by, f(x) = 1 (2π)p/2√det Σ exp −1 2(x−µ)TΣ−1(x−µ)(3.1) where µ=hxiis the mean vector and Σ is the covariance matrix, whose elements are given by Σab =h(xa−µa)(xb−µb)i,(3.2) with a, b = 1, . . . , p. Let us now consider the following decomposition of the random vector x= (x1,x2), where x2are pccomponents of the vector xthat will be constrained by a condition x2=˜x, 2Note, however, that this covariance function leads to a somewhat special form of the Hessian matrix for the minima in this GRF (See for example the discussion of this point in [18].) It would be interesting to check whether this could have any quantitative effect on the conclusions of our paper. 3See a brief description of the Kac-Rice formula in the current context in appendix A.5. – 4 – JHEP05(2020)142 and x1are the remaining p−pcunconstrained elements. Then one can show [30,31] that the distribution probability for x1holding x2fixed to the desired values is given by, ˜ f(x1|x2=˜ x) = 1 (2π)p−pc 2pdet ˜ Σ exp −1 2(x1−˜ µ)T˜ Σ−1(x1−˜ µ),(3.3) which shows that the distribution for the variables x1is indeed a Gaussian distribution but now with a mean and covariance functions given in terms of the original ones as ˜µ=µ1+ Σ12Σ−1 22 (˜x−µ2),˜ Σ=Σ11 −Σ12Σ−1 22 Σ21 ,(3.4) where µ1and µ2are the means of the vectors x1and x2respectively, and Σ11 =h(x1−µ1)(x1−µ1)i, Σ12 = Σ21 =h(x1−µ1)(x2−µ2)i, Σ22 =h(x2−µ2)(x2−µ2)i.(3.5) This is possible because one can always find a new Gaussian random vector x0= (x0 1,x0 2), connected to the original one with a non-singular linear transformation x0=A·x, such that x0 2=x2is uncorrelated to x0 1. We show in appendix A.2 a proof of this statement. In the rest of the paper we will use this fact in several different ways, applying this technique for Gaussian random vectors made of different quantities of our potential. 3.1 Slepian models for critical points In this section we will use the methods described earlier to generate a Gaussian random field with a critical point with a specific height at the center, φ=0. In other words, we will find a description of the new GRF conditioned so that the point at its center satisfies the following properties: V(0) = uand V0 i(0) = ηi= 0 for i= 1,2. In order to do this we will follow the prescription used in the mathematical literature for maxima in GRF [32] and adapt it to our case. Let us start by introducing the following Gaussian random vector: x={V(φ1), . . . , V (φq), V (0), η1, η2, ζ11, ζ22, ζ12}(3.6) where we denote by φa, with a= 1, . . . , q, the position in field space of a discrete set of qpoints. One can show that the Gaussian random vector xhas zero mean, and a probability distribution that can be readily computed using the form of the covariance function and its derivatives. This is a somewhat lengthy calculation and we have given the general expression in appendix A.6. According to the description for constrained Gaussian random vectors given above this is all we need to obtain the new mean and covariance function for the new conditioned vector (and thus, also for the constrained GRF). Using the results in appendix A.6, one can show that the new mean function for the GRF with the constrained conditions is given by, ˜µ(φ) = e−φ2 2Λ2"u1 + φ2 2Λ2+1 2 2 X i=1 φ2 iλi#.(3.7) – 5 – JHEP05(2020)142 This result corresponds to the particular choice of covariance function in eq. (2.3), and is written in terms of the value of the field V(0) = uand the eigenvalues of the Hessian matrix at the center, λi, which are to be drawn from the distribution in eq. (3.9) below. The new covariance function is ˜ C(φ1,φ2) = U2 0exp −|φ1|2+|φ2|2 2Λ2exp φ1·φ2 Λ2−1−φ1·φ2 Λ2−(φ1·φ2)2 2Λ4, (3.8) which is no longer homogeneous, but it is still isotropic. It is important to note that the eigenvalues of the Hessian are not statistically independent of the height of the potential. This is intuitively clear since, for example, one would expect the typical minimum at a large height to be quite shallow compared to the minima situated well bellow the mean value of the potential. This expectation can be translated to the existence of important correlations between the field and its second derivatives at a point, and in particular at critical points. In order to take this effect into account one can calculate the joint probability distributions for the Hessian eigenvalues (λi) and heights (u) at critical points to obtain4 Pu,λ du Y i dλi=Nexp −u2 2U2 0|λ1−λ2| 2 Y i=1 |λi|exp "−Λ2λi+u 2U02#dλidu , (3.9) where Nis a normalizing constant. This distribution includes all types of critical points, namely maxima, minima and saddle points. Depending on the kind we are interested in, we simply need to impose positivity or negativity conditions on the values of each λi. Using these results we can generate a Gaussian random field with a critical point with the desired properties by the following procedure. Let us consider for example a minimum with fixed height u. Our first step will be to generate a set of eigenvalues drawn from the distribution (3.9) taking into account the value of u, imposing the non-negativity condition λi≥0, and fixing the normalization factor accordingly. Using these values for λiwe can then generate realizations of the potential using the expression V(φ) = e−φ2 2Λ2"u1 + φ2 2Λ2+1 2 2 X i=1 φ2 iλi#+ ∆(φ) (3.10) where we have denoted by ∆(φ) an inhomogeneous, zero-mean Gaussian random field whose covariance function is given by ˜ C(φ1,φ2) in eq. (3.8). We show in figure 1an example of the different ingredients that make up a Slepian model for a local minimum in a 1dGRF. We can use a similar procedure to generate other critical points, such as saddle points with different number of negative eigenvalues, by generating the appropriate samples of λi’s. An important conclusion that can be derived from the Slepian model (3.10), first noticed in [32], is that for highly non-generic extrema |u|  U0(such as very low maxima or high minima), the shape of this GRF becomes very deterministic around the critical 4See the calculation in appendix A.6. – 6 – JHEP05(2020)142 μ(ϕ) Δ(ϕ) -4-2 2 4 ϕ -2.0 -1.5 -1.0 -0.5 0.5 -4-2 2 4 ϕ -2.0 -1.5 -1.0 -0.5 0.5 V(ϕ) Figure 1. A 1d example of a Slepian model of a constrained minimum in a GRF. We show, for a particular realization, the two separate components of the construction on the left, namely, the constrained mean field form µ(φ) in eq. (3.7) and the inhomogenous new GRF ∆(φ) with covariance function given by eq. (3.8). The total GRF is shown on the right. point, and it is described very accurately by the first two terms in eq. (3.10). Indeed, one can see from eq. (3.8) that the standard deviation of the random component ∆(φ) is always smaller than U0, and that it approaches zero near the extremum located at φ=0(see also figure 1). Therefore the last contribution in (3.10) can be neglected in a neighbourhood of the extremum where |∆(φ)|.U0 |V(0)|holds. On the other hand, in the limit |u|  U0the eigenvalue distribution of the Hessian (3.9) is approximately given by5 Pλdλ1dλ2∼ |λ1−λ2||λ1||λ2|exp −Λ2|(λ1+λ2)u| 2U2 0dλ1dλ2,(3.11) which indicates that in this limit the magnitude of the eigenvalues is very suppressed |λi|  U0/Λ2. Then, as we mentioned above, for highly non-generic extrema the decomposition (3.10) is dominated by its deterministic part (the first term), what makes these Slepian models very predictive in those situations. As we shall see bellow, this result is particularly important when we consider the distribution of non-perturbative decay rates from minima with a large vacuum energy. For an example of a realization with a high minimum see figure 2(a). This deterministic character of large fluctuations of Gaussian Random Fields plays an important role in various areas of Cosmology, such as the analysis of the CMB data, and the study of Large Scale Structure formation in the universe (see e.g. [33–37]). 3.2 Slepian models for inflection points As we discussed in the Introduction, we are also interested in inflection points in the landscape. The reason is that in a cosmological context these points could be one of the regions of the potential that give rise to a cosmological inflationary period. However, in order to be compatible with the latest cosmological observations, one needs to restrict the form of these inflection points. This leads us to consider an inflection point at φ= 0 as a realization of the GRF with a small gradient of the potential in the φ1direction, denoted by η1, and the rest of the coefficients of the Taylor expansion of the field around that point 5Note that for very high minima u > 0 and λi>0, while for very low maxima all signs are reversed. – 7 – JHEP05(2020)142 ▲ ▲ ▲ ▲ ▲▲▲▲ ■ ■ ■ ■■■■■ Optimal path ▲ Linear approximation ■Thin-wall approximation -2-1 0 1 2 3 4 5 V(ϕFV)/U0 100 103 104 105 S Figure 6. Evolution of the median of the action, with error bars representing data between the first and third quartiles of each distribution, for the optimal path, the linear (straight-path) and the thin wall approximations, along with a fitting curve (see (4.6)). of (4.6) for the expected value of the action. This enhancement of the predictability of the Slepian model for large values of Vfv corresponds precisely to what we anticipated in the previous section. Indeed, there we showed that near high minima the random potential becomes dominated by the first term in the decomposition (3.10), and therefore the landscape is very deterministic in a neighbourhood of false vacua with large Vfv. Consistent with this result, when studying the non-perturbative stability from these vacua we observe a reduction of the variance of tunneling actions for large heights of the false vacuum. This agreement also suggests that in the case of minima with a large Vfv the value of the instanton action is dominated by the local structure of the minimum. We will provide further evidence for this claim below. 4.2 Approximations for the calculation of the action Due to the inherent instability of the equations to be solved to compute tunneling profiles, it is clear that as we increase the domain and dimensionality of the potential under study, the required computational time to solve the system will grow accordingly. Evidently, this makes the study of higher-dimensional GRFs and their tunneling properties almost prohibitive in this sense. Motivated by these limitations, we turn to computing several different approximations of tunneling actions suggested in the literature, and compare them with our exact results. 4.2.1 Thin wall approximation The thin-wall prescription was already discussed in the original papers by Coleman in [21]. In this approximation the instanton action is given in terms of the difference between potential at the false vacuum (Vfv) and true vacuum (Vtv) and σ, the tension of the wall – 14 – JHEP05(2020)142 interpolating between them, namely, ¯ Stw =27π2σ4 2(Vfv −Vtv)3, σ =ZφF V φT V dφ p2(V(φ)−V(φTV )) .(4.7) This approximation is accurate as long as the difference between Vfv and Vtv is small. We evaluated (4.7) for each bounce we previously found with AnyBubble in order to check this expression and its predictive power for GRFs. In the computation we restricted the field to a straight line in field space connecting the true and false vacua. Figure 6shows the evolution of the median of ¯ Stw as a function of the false-vacuum height. While the width and median of the distribution in this case follow the same pattern as the optimal action, the values diverge rapidly from the optimal ones as the false vacuum height increases. This is not too surprising since, as one increases the height of the false vacuum minimum, the field can tunnel to a minimum with quite different values of the potential, what violates one of the premises of the thin wall approximation. 4.2.2 Straight-path approximation While the thin-wall prescription provides a solid upper bound on the bounce action [43], it does not provide any useful estimation on the actual value on the bounce in our case. This fact calls for an alternative way to estimate the action, mostly for higher-dimensional landscapes. A straightforward simplification to this problem was introduced in [44], which we will denote by straight-path approximation. This prescription is based on reducing the field space to a single straight line connecting the false and true vacua, thus making the problem of tunneling effectively one-dimensional. As can be seen from figure 3, this approximation may not be too unreasonable. Even though there are some paths which do curve over the field space, many (if not most) of them follow a straight trajectory in field space. Note, however, that this restriction in field space may yield effective potentials where the bounce does not exist or might even correspond to a different bounce in the full theory. For more details on the properties of this approximation, see [45]. For each optimal path, we considered a straight line in the two-dimensional GRF connecting the true and false vacua, and computed the corresponding estimate of the action, ¯ Ssp, in each case. In principle, ¯ Ssp represents an upper bound on the optimal action ¯ S, as the former only considers variations of the action in the direction of the straight path [44]. It is thus expected (and explicitly shown in [45]) that this approximation will diverge from the full solution as the dimensionality of the potential is increased. We found that in this case the distribution of actions in terms of false vacuum height is identical to the optimal one shown in figure 5, though slightly shifted to higher values. As we can see from figure 6, the change in the median is minimal when the straight-path approximation is considered. Although, as we just mentioned, the straight-path approximation is not expected to give precise results for potentials in a higher field space dimension, this result suggests that it would be interesting to explore the validity of this method with GRFs in higher dimensions. Indeed, due to the computational complexity of such an anal- – 15 – JHEP05(2020)142 FV height -2-1 0 1 2 3 4 5 0π/8π/4 3π/8π/2θ 0.0 0.2 0.4 0.6 0.8 1.0 Pθ Figure 7. Exit angle distribution with respect to direction of the lowest eigenvalue for the instanton path of the most probable decay channel in each generated potential. ysis, a rough statistical estimate of the decay rate obtained with this approximation would still be very valuable. 4.3 The lowest action In many circumstances one will be interested in the lowest action for a particular kind of minima. This will of course correspond to the path that would dominate the decay for those minima. In this subsection we will investigate the characteristics of such trajectories in field space. 4.3.1 Exit angle An intuitive way to think about the most likely decay process would be to imagine that the tunneling occurs along the trajectory with the lowest barrier. One can check this idea in our case by first identifying the angle (in our 2dfield space), θ, that the instanton trajectory makes with respect to the direction of the lowest eigenvalue of the Hessian at the minimum. A distribution of such angles obtained for different values of the height is plotted in figure 7. We see that there is a clear tendency of the tunnelings to occur around θ≈0 but the correlation is not very strong. 4.3.2 Estimating the lowest action The correlation of the instanton path with the lowest eigenvalue direction at the false vacuum suggests that one can try to estimate the lowest action by analyzing the potential along the lowest eigenvalue direction alone. This has been recently proposed in the context of the landscape in [26]. In the following we will use our large sample of realizations to test this idea in detail in our 2dGRF model of the landscape. – 16 – JHEP05(2020)142 ◆ ◆ ◆ ◆ ◆◆◆◆ Min. action per potential ◆Sarid app. along lowest eig. -2-1 0 1 2 3 4 5 V(ϕFV)/U0 1 100 103 104 105 S Figure 8. Distribution of lowest action per potential and Sarid approximation [46] along the lowest barrier direction, in terms of false vacuum height. The fit in eq. (4.6) is shown for comparison with previous results. In order to evaluate the instanton action along the lowest eigenvalue direction we first take a slice of the potential along that direction and fit it to be of the form, Vle(φ1) = V0+1 2λ1φ2 1+1 3!ρ111φ3 1+1 4!δφ4 1.(4.8) Note that this procedure does not guarantee that the resulting one-dimensional potential is suitable for a tunneling process. In fact, in many cases the potential constructed this way does not have a lower minimum along this direction and therefore it cannot be used to estimate the decay rate. In the following we will only compute the instanton action in the successful cases where this 1dtruncation gives an acceptable form, what in particular requires ρ111 <0. Considering this simple form of the potential as the most likely exit path for the decay transition we can estimate the instanton action. In order to do that we will use the parametrization of the Euclidean action for the bounce that was obtained by Sarid in [46]. In our notation this becomes, ¯ SS=   18λ1 ρ111245.4−46.1 + 2π2 12(1−4κ)3+16.5 (1−4κ)2+28 1−4κ, κ > 0 18λ1 ρ111245.41+(136.2 2π2)1.1|κ|1.1−1/1.1, κ ≤0 (4.9) where κ=3 4δλ1 ρ2 111 .(4.10) We show in figure 8the distributions of the lowest action from the exact computation and compare it to this estimate along the lowest barrier direction. We notice that the agreement between these two results is pretty good, what suggest that one can use this – 17 – JHEP05(2020)142 approximation to estimate the decay rate of vacua in a Gaussian random landscape. Moreover, it is worth noting that this approximation depends only on the local structure of the minimum, precisely where the Slepian model has a large predictive power for large values of Vfv. The expression (4.9) becomes increasingly accurate for large values of the false vacuum energy Vfv, what indicates that in this regime instanton action is mostly determined by the local form of the minimum. On the other hand, according to the Slepian model, the scalar potential around all high minima should look very similar in all realizations, with its shape dominated by the first term in (3.10). This explains why the distribution of instanton actions becomes more deterministic (figure 5) for larger values of Vfv, and therefore also the agreement between the Sarid approximation (4.9) for the lowest action and our fit in eq. (4.6)for the median of the distribution. It would be interesting to check if this good agreement persists on a much larger landscape with hundreds of directions in field space,10 and whether the approximation (4.9) can be used in combination with our Slepian model make robust predictions regarding the tunneling rates of high vacua. 5 Inflation in a Slepian random landscape Up to now we have been using all the software and mathematical tools described above for the computation of bounce profiles and actions with Gaussian random fields conditioned to have a minimum at φ=0. In this section, we turn to studying constrained GRFs with inflection points at the origin of field space focusing on their application to cosmological inflation. Inflation in random potentials has already been extensively studied [13,14,19,20,47]. More specifically, inflation around inflection points has received special attention for being capable of sustaining enough e-folds to make contact with observations, while taking place in a small region of field space with an effectively one-dimensional potential. While most of the obtained results and distributions seem promising, they have only been tested within Taylor expansions around these points, instead of using full GRFs. As we mentioned before, such methods do not capture correctly the global features of the potential, what is essential for characterising the non-perturbative stability of vacua. Therefore, this procedure is unsuitable for studying models of inflation where the initial conditions are determined by the decay of a parent false vacuum. In this section we will apply Slepian models to constrain Gaussian random fields to have inflection point with the desired properties to sustain inflation, and then we will study the dependence of its cosmological observables on the initial conditions, set by different realizations of the parent vacuum. 10Note that in our calculation we kept the quartic term of the potential while in reference [26] the authors drop this term arguing that for large number of fields (N) this coefficient becomes irrelevant. We have checked that in our case this is not the case and in order to obtain a good agreement it is necessary to take this term into account. This is due to the fact that we have limited our investigation to the N=2 case. – 18 – JHEP05(2020)142 5.1 1D inflection point inflation Let us briefly review the main results for one-dimensional inflection-point inflation (see [20,48] and references therein for more details). Let us consider a potential of the form, V(φ) = u+ηφ +1 6ρφ3,(5.1) where, in order to satisfy the slow-roll conditions around the inflection point, we will assume that ηu. Note that we do not need to assume that the third derivative is too small. In fact, following typical conditions for a GRF we will consider the case where uρ. Taking this into account one can show that slow-roll inflation conditions will be satisfied in the interval −u ρ<φ<u ρ,(5.2) which together with the condition uρimplies that we are describing small field inflation. Using the slow-roll conditions, it is easy to check that the expected number of e-folds, Nexp, that can be achieved within that region is Nexp =Zu/ρ −u/ρ dφ √2≈π√2u √ηρ −4≡Nmax −4,(5.3) where = (V00(φ)/√2V(φ))2and Nmax is the maximal number of e-folds achievable in the whole potential. Moreover, defining x≡πNCMB Nmax , y ≡Nmax 2π,(5.4) where NCMB is the e-fold number at which the CMB scales leave the horizon, the spectral index of scalar perturbations can be shown to be given by ns= 1 + 2 ytan x−y 1 + ytan x.(5.5) Finally, the amplitude of scalar perturbations can be expressed as ∆2 R=1 12π2 V3(φ) V0(φ)2≈N4 CMBρ2 48π2uf2(x, y) (5.6) where f(x, y) = cos2(x)(ytan(x) + 1)2 x2(y2+ 1) ,(5.7) satisfies f(x, y)∼1 for y1 and x∼1. With these expressions at hand, we can easily obtain a set of parameters for the inflection point (u, η and ρ) that are in agreement with the current cosmological observations, namely, Nexp > NCMB ≈50, ns≈0.965 and ∆2 R≈2×10−9(see eq. (5.8) below). – 19 – JHEP05(2020)142 5.2 Numerical inflection points in a 2D landscape We now want to embed 1dinflection-point inflation in our 2dGRF landscape. In order to do that we can follow the procedure explained in section 3.2 for Slepian models in the case of inflection points. In the notation introduced earlier, the 1dparameters η=η1and ρ=ρ111, correspond to the derivatives along the flat direction of the multidimensional inflection point. Note that, in principle, uand ρ111 (when evaluated at the same point) are uncorrelated, but the same is not true for uand the second derivative along the inflaton direction λ1; similarly η1and ρ111 are also correlated, see eq. (3.17). Here we are interested in studying the global properties of the landscape on the cosmological observables so we will focus on a particular type of inflection point where we have fixed its 1dparameters.11 Following the steps from the previous section, we built two-dimensional GRFs with an inflection point whose inflating direction has fixed features. In the forthcoming sections we set u= 0.5U0, η1= 6.8·10−6U0 Λ, ρ111 = 2.5U0 Λ3(5.8) where U0= 6.0·10−16 M4 Pl and Λ = 0.5MPl define the energy scale and correlation length respectively, with the Planck masses written explicitly for clarity. Once u,ηand ρhave been fixed, using the probability distributions listed in (3.16) and (3.17), we can obtain the remaining parameters of the two-dimensional inflection point set at the origin of field space φ=0, and generate in a efficient way a large sample of GRFs with the listed properties.12 As an example, we show in figure 9a field constructed with the above constraints. We then used AnyBubble to tunnel from a higher false vacuum to the central inflection point. We note that even though in every realization the inflection point has the same properties along the φ1direction up to third order, the potentials are different away from that point. This means that the false vacuum, which decays to the region around the inflection point, is located in a different place and it also has different features in each realization, e.g. vacuum energy and barrier height. Using AnyBubble we computed the exit points of a large set of realizations. After that we used these exit points of the instanton decay as the starting points of a Lorentzian evolution of a FRW universe with this potential. In order to study the inflationary trajectory we used mTransport [49], a Mathematica code developed to compute inflationary observables using the transport method. The cosmological evolution inside of a bubble universe created from tunneling is described by an open FRW universe [50]. Here, for simplicity, we used the flat-space approximation for the evolution of the cosmological interior of the bubble.13 11It is also interesting to study the effects of varying these parameters together with the global properties of the GRF. We leave the details of this calculation for a future publication. 12Note that following our earlier definition of the inflection point in our 2dlandscape, we have set η2= 0 and λ2>0. 13Note that in reality the initial cosmological evolution is dominated by the spatial curvature of the open FRW slices that describe the bubble interior. This will have some effect on the initial stages of the evolution of the scalar field in a multidimensional potential. See [19,28] for a discussion of these effects. – 20 – JHEP05(2020)142 Figure 9. A Gaussian random field conditioned to have an inflection point in the middle. The dashed line represents the tunneling from a minimum to a lower inflection point. The inflationary slow-roll phase starts at the exit point, inflates for around 124 e-folds following the solid line, and evolves towards the closest minimum. We only show the inflationary part of the trajectory. Green, yellow and red dots represent minima, saddle points and maxima of the potential. The inflection point is marked with a blue dot. In the example from figure 9, the dashed line represents the tunneling trajectory, while the solid one marks the inflationary one. We found this path to sustain a total of 124.1 e-folds and a spectral index of ns= 0.964 at the observable scale. 5.3 Statistics of inflationary parameters In order to test the method described above to generate inflationary random fields, we generated 5000 GRFs constrained to have an inflection point with the same properties as the one in the example of figure 9(see eq. (5.8)). Next, in each of these realizations, we found all minima lying above the central inflection point and used anyBubble to compute the tunneling trajectory from the former to the latter in each case. Considering the exit point as the starting point of an inflationary phase, we used mTransport to find the number of e-folds, power spectrum, tensor-to-scalar ratio, spectral index and its running. The distributions of the e-fold number and the spectral index are shown in figure 10, for a pivot scale of 50 e-folds, whereas the action associated to the tunneling to the inflection point is shown in figure 11. This is a different distribution than the ones we found earlier, since the common factor in these decays is the final point and we do not impose anything about the initial (false vacuum) state. It is interesting to see that this distribution is quite peaked around an action of the order of 103. – 21 – JHEP05(2020)142 110 120 130 140 150Ne 0.00 0.05 0.10 0.15 0.20 0.25 PNe (a) 0.960 0.965 0.970 0.975 ns 0 100 200 300 400 Pns (b) Figure 10. (a) Distribution of number of e-folds, with Nexp shown with a dashed line (b) Histogram of the obtained spectral index, with the analytic prediction marked with a dashed line. Both figures represent 4000 inflationary trajectories (see text). We have also obtained the distributions for the amplitude of scalar perturbations, tensor-to-scalar ratio and running of spectral index which turned out the be centered around the values ∆2 R= (2.02±0.04)·10−9, r = (8.0±0.1)·10−9and α= (−2.49±0.02)·10−3,(5.9) respectively.14 Our results in this section are fully compatible with the 1dstudies in [14]. Finally, in figure 12, we show several inflationary trajectories corresponding to tunnelings in different GRFs with an inflection point in the middle with the same features. Note that all trajectories, no matter how far they start from, have a similar behavior. After oscillating in the vertical φ2direction, they all stabilize around the inflection point and inflate along it. Most of the e-folds happen in the vicinity of the inflection point, as predicted by the analytic estimation. We have obtained successful results from this analysis around 80% of the times. The rest of the times the procedure did not yield a cosmological solution in agreement with our universe either because inflation ended too soon or because the exit point was too far from the central inflection point and the inflaton trajectory went astray. The successful paths show very good agreement with the 1dresults presented in the previous section. We see that even though some of the trajectories have some substantial deviation from the 1d inflationary direction, the cosmological observables are still in pretty good agreement with the single field inflection point inflation. The distributions of the results are quite peaked around their central values, so we can conclude that the dependence of the observables on the initial conditions seems to be quite mild. 14The cosmological evolution of these Lorentzian trajectories continue after inflation until they reach a lower minimum. We have not fine-tuned this minimum to be in Minkowski space, so in general the evolution leads to eternal de Sitter or to an Anti-deSitter crunch. We are only interested in the statistics of the inflationary period so we have stopped this evolution after the field leaves the slow-roll regime. – 22 – JHEP05(2020)142 2 4 6 8 Log10[S] 0.0 0.1 0.2 0.3 0.4 PS _ Figure 11. Distribution of the tunneling action from a minimum to the central inflection point, right before inflation begins. 0.05 0.10 0.15 0.20 ϕ1 -0.2 -0.1 0.1 ϕ2 Figure 12. Showcase of several inflationary trajectories from different tunnelings to the central inflection point. Each exit point is marked by a blue dot. It is important to remember that all these realizations have the same 1dinflection point parameters. In order to extract the complete statistical information about the predictions of a particular GRF we should combine these results with the ones obtained from inflection points with other parameters with their correct statistical weight. This is a much more numerically intensive problem and we leave it for a future publication. – 23 – JHEP05(2020)142 Let us change the notation to ∂φjV(0) = V0 j(0) and evaluate the previous expression for some useful cases: hV(0)V(0)i=U2 0(A.25) hV(0)V0 i(0)i=V0 i(0)V00 jk(0)= 0 (A.26) V0 i(0)V0 j(0)=−V(0)V00 ij (0)=−∂2C(0) ∂φi∂φj =α2δij (A.27) V00 ij (0)V00 kl(0)=∂4C(0) ∂φi∂φj∂φk∂φl =   α22 if i=j6=k=l(and perms.) α4if i=j=k=l 0 otherwise. (A.28) V(0)V000 jkl(0)=V00 ij (0)V000 klm(0)= 0 (A.29) V0 i(0)V000 jkl(0)=−V00 ij (0)V00 kl(0)=   −α22 if i=j6=k=l(and perms.) −α4if i=j=k=l 0 otherwise. (A.30) V000 ijk(0)V000 lmn(0)=−∂6C(0) ∂φi∂φj∂φk∂φl∂φm∂φn =       α222 if i=j6=k=l6=m=n(and perms.) α24 if i=j6=k=l=m=n(and perms.) α6if i=j=k=l=m=n 0 otherwise. (A.31) In the above expressions, αi,αij and αijk are numerical constants which depend only on the covariance function of the (unconstrained) Gaussian random field. Note that in the two-dimensional case α222 will be absent from all derivations, since the indices appearing in the correlation function between the third derivatives can only take two different values. Note also that odd derivatives of the GRF are uncorrelated with even ones when they are evaluated at the same point in field space. This is due to the isotropy of the covariance function: if it is written as a power series, only even powers such as φ2 i, φ2 iφ2 jwill be involved. Therefore, only those correlations which end up involving even derivatives of the covariance function are non-zero. This however, does not mean the fields V(φ) and, say, V0 i(φ) are completely uncorrelated. If we evaluate them at different points in field space, it can be shown [30, theorem 2.3] that V(φ)V0 i(0)=−∂ ∂φi C(φ) (A.32) V(φ)V00 ij (0)=∂2 ∂φi∂φj C(φ) (A.33) V(φ)V000 ijk(0)=−∂3 ∂φi∂φj∂φk C(φ) (A.34) therefore, a GRF and any of its derivatives are correlated as processes. – 30 – JHEP05(2020)142 A.5 The Kac-Rice formula and conditioned Gaussian random fields Consider a Gaussian random vector field with components V(φ) = {V1(φ), . . . , Vn(φ)}. The multidimensional16 Kac-Rice formula for this field gives us the expected number of times a certain event, say, V(φ) = u, happens in an interval φ∈Iof volume V: E#,I [V(φ) = u] = ZI dφ|det V0(φ)|δ(V(φ)−u)(A.35) where det V0(φ) stands for the Jacobian determinant of the vector field,17 that is, V0(φ) =    ∂φ1V1(φ)··· ∂φ1Vn(φ) . . .. . . ∂φnV1(φ)··· ∂φnVn(φ)   .(A.36) If the field is stationary, that is, homogeneous and isotropic, we can simplify the expression above. Denoting V0=V(0) and V0 0=V0(0), we find, assuming ergodicity, E#,I [V(φ) = u] = VZdV0dV00|det V00|δ(V0−u)P(V0,V00) (A.37) where the integral is performed over the whole domain of V0and V0 0and P(V0,V00) is the joint PDF of V0and its derivatives. More than one simultaneous event can be considered in the expressions above by enlarging the vector Vand introducing more Dirac deltas representing each event.18 While the above expression can certainly be used to obtain the number of times a certain event happens in a given interval, it can also be used to obtain distribution functions. More specifically, applying ergodicity theorems, it can be shown [30] that the probability of an event Ahappening, given that Bhas happened, that is, P(A|B), can be obtained by P(A|B) = E#,I [A∩B] E#,I [B].(A.38) If Adepends on continuous parameters (such as the position in field space of the GRF), then the expression above represents a probability distribution function. A.6 Conditioned Gaussian random field for a critical point With the tools presented in the sections above, we are now ready to begin conditioning GRFs. We can begin applying (A.38) and specializing it for critical points. We denote by Athe event describing the field V(φ) taking a particular configuration, while Bimposes V(0)≡V0=uand V0 i(0)≡ηi= 0, that is, a critical point lying in the center of field space at height u. In order to proceed more easily, we shall discretize V(φ) as {V(φ1), . . . , V (φq)}≡{V1,...Vq} ≡ V. 16Note that this formula is only valid for fields mapping Rn→Rn. 17For critical points, the Jacobian is identical to the Hessian of the GRF at the critical point. 18See, however, [30, ch.8] for a discussion on different types of conditioning events and how to deal with them. The reason why we consider the V0=uevent simply with a Dirac delta is that it is a vertical window conditioning event. – 31 – JHEP05(2020)142 In this case, the conditioning event involves the Gaussian random vector field V=∇V, whose Jacobian is the Hessian of the original field Vevaluated at φ=0. Therefore, its determinant is simply the product of the eigenvalues of the Hessian evaluated at the origin, Qn i=1 λi. Applying the Kac-Rice formula (A.37) into (A.38) yields PV(φ)V0=u,∇V0=0≡Pcp[V(φ)] = =Zn Y i=1 dηiδ(ηi)dλi|λi|∆(λ)δ(V0−u) q Y j=1 d˜ Vjδ(˜ Vj−Vj)PV0,V,η,λ Zn Y i=1 dηiδ(ηi)dλi|λi|∆(λ)δ(V0−u)PV0,η,λ(A.39) =NZn Y i=1 (dλi|λi|)∆(λ)PV(φ),λ1,...,λnV0=u, ∇V0=0(A.40) where the integration domain will depend on the kind of critical point we are working with. ∆(λ)∝Qi<j |λi−λj|is the Jacobian of the variable change from components of the Hessian matrix to its eigenvalues, the proportionality constant depending on the dimensionality of the field space. For simplicity, the denominator in (A.39) has been considered as a normalization factor for the distribution in the numerator. We can rewrite (A.39) in a more useful way: Pcp[V(φ)] = Y iZdλiqu(λ1, . . . , λn)PV(t)V0=u, ∇V0=0, λ1, . . . , λn(A.41) where qu(λ1, . . . , λn) = Y i|λi|∆(λ)Pλ1, . . . , λnV0=u, ∇V0=0(A.42) represents the distribution of the Hessian eigenvalues at the origin for a critical point of height u. However, due to the homogeneous and isotropic nature of the original GRF, the latter distribution is valid for any critical point in the GRF, thus giving us a distribution for the parameters at critical points in the unconstrained field. Equations (A.41) and (A.42) are central results in this derivation. Note that the Qi|λi|∆(λ) factor is a direct consequence of the Kac-Rice formula, and as we shall explicitly see in appendix B, it carries important consequences in the distribution of the eigenvalues at critical points. We can now see the power of this method. Assuming we have discretized our field space, we can readily compute the conditional probability distributions in (A.41) and (A.42) using the results from section A.2. This leads, together with (A.42), to a distribution from which we can draw eigenvalues for a minimum of height u. These can be plugged in (A.41) to generate iterations of GRFs with a minimum (or any other critical point) at their origin. In order to apply all this machinery, let us introduce the following Gaussian random vector: {V(φ1), . . . , V (φq), V (0), V 0 1(0), . . . , V 0 n(0), V 00 11(0), . . . , V 00 nn(0), V 00 12(0), . . . , V 00 (n−1)n(0) | {z } V00 ij (0)i<j } (A.43) – 32 – JHEP05(2020)142 where we denote by φqthe position in field space of a discrete set of points whose center is located at 0,V0 i(0) describes the first derivative along φiand V00 ij (0) is the (i, j)-th element of the Hessian matrix. In order to unclutter the notation, we will compactify the previous vector as {V, V (0),V0(0),V00(0)}(A.44) which has dimension q+1+n+n+1 2n(n−1). The mean of (A.43) is zero, and the covariance matrix of these quantities can be computed from the results in section A.4: Σ =      SV V SV0SV1SV2 S0VU2 00S02 S1V0S11 0 S2VS20 0S22      (A.45) where S02 =−α2··· −α20··· 0=ST 20 (A.46) S11 =α2×1n(A.47) S22 =              α4α22 ··· α22 α22 α4··· α22 0 . . .. . ..... . . α22 α22 ··· α4 α22 0 0... 0α22              (A.48) SV V =      C(0)C(φ1−φ2)··· C(φ1−φq) C(φ2−φ1)C(0)··· C(φ2−φq) . . .. . ..... . . C(φq−φ1)C(φq−φ2)··· C(0)       (A.49) S0V=C(φ1)C(φ2)··· C(φq)=ST V0(A.50) S1V=      −C0 1(φ1)−C0 1(φ2)··· −C0 1(φq) −C0 2(φ1)−C0 2(φ2)··· −C0 2(φq) . . .. . ..... . . −C0 n(φ1)−C0 n(φ2)··· −C0 n(φq)       =ST V1(A.51) S2V=            C00 11(φ1)··· C00 11(φq) . . ..... . . C00 nn(φ1)··· C00 nn(φq) C00 12(φ1)··· C00 12(φq) . . ..... . . C00 (n−1)n(φ1)··· C00 (n−1)n(φq)            =ST V2(A.52) – 33 – JHEP05(2020)142 In order to simplify the notation, since the jointly Gaussian probability distribution in the end depends on two-point functions, we can actually write19 (A.45) in the following way: Σ =        U2 0C(φ1−φ2)C(φ1)SV1(φ1)SV2(φ1) C(φ2−φ1)U2 0C(φ2)SV1(φ2)SV2(φ2) C(φ1)C(φ2)U2 00S02 S1V(φ1)S1V(φ2)0S11 0 S2V(φ1)S2V(φ2)S20 0S22        (A.53) where SV1(φ) = −C0 1(φ)··· −C0 n(φ)=ST 1V(A.54) SV2(φ) = C00 11(φ)··· C00 nn(φ)C00 12(φ)··· C00 (n−1)n(φ)=ST 2V(A.55) With these arrangements, the Gaussian random vector corresponding to (A.53) is V(φ1), V (φ2), V (0),V0(0),V00(0).(A.56) We have decomposed (A.53) into blocks so it can be plugged into (A.57) and (A.58) to obtain the mean function and covariance matrix of the conditioned process.20 Using the results given above, one gets that the expectation value for the GRF around a critical point where V0=uand V0 0= 0, is given by, ˜µ(φ) = µ(φ) + C(φ)SV1(φ)SV2(φ)   U2 00S02 0S11 0 S20 0S22    −1   u 0 h    =C(φ)SV2(φ) U2 0S02 S20 S22 !−1 u h!(A.57) where h=h11, . . . , hnn, h12, . . . , h(n−1)nrepresents a certain configuration of the Hessian components of the field around the origin. Furthermore, the covariance function for the conditioned GRF is now ˜ C(φ1,φ2) = C(φ1−φ2)−C(φ1)SV1(φ1)SV2(φ1)   U2 00S02 0S11 0 S20 0S22    −1   C(φ2) S1V(φ2) S2V(φ2)   =C(φ1−φ2)−C(φ1)SV2(φ1) U2 0S02 S20 S22 !−1 C(s) S2V(φ2)! −SV1(φ1)S−1 11 S1V(φ2) (A.58) 19We basically have evaluated the first row for a given φ1and the first column for a given φ2, just as in [32]. Doing so allows us to treat the independent variable as a continuous one, rather than a discrete one. 20Strictly speaking, we should be getting the mean and covariance of the random vector {V(φ1), V (φ2)}. Due to the isotropy of the GRF, φ1and φ2can be any points in field space. Thus, in order to unclutter the notation, we will only keep track of a single component of the resulting mean vector. Likewise, we will only keep the hV(φ1)V(φ2)icomponent of the covariance matrix. – 34 – JHEP05(2020)142 We can also obtain (A.42), the distribution of eigenvalues at a critical point of a given height u, following the same steps as above, using as initial covariance matrix the bottomright block of (A.53). A.6.1 Analysis of a conditioned 2D Gaussian field Let us apply these expressions to a two-dimensional isotropic and homogeneous GRF with covariance function C(φ) = U2 0exp −φ2 2Λ2,(A.59) and zero mean. For this case, we obtain the conditioned mean from (A.57), which gives ˜µ(φ) = e−φ2 2Λ2"u1 + φ2 2Λ2+1 2φ1φ2 h11 h12 h21 h22 ! φ1 φ2!#,(A.60) where h21 =h12, by definition. Since we are free to choose the basis of φ, in order to simplify the expression we will employ the eigenvector basis of the Hessian matrix, therefore transforming (A.60) to ˜µ(φ) = e−φ2 2Λ2"u1 + φ2 2Λ2+1 2 2 X i=1 λiφ2 i#,(A.61) where λidenote the two eigenvectors, drawn from (A.42) specialized to this case (see below). As for the conditioned covariance, from (A.58) we obtain ˜ C(φ1,φ2) = U2 0exp−|φ1|2+|φ2|2 2Λ2expφ1·φ2 Λ2−1−φ1·φ2 Λ2−(φ1·φ2)2 2Λ4.(A.62) Note that the covariance function of the conditioned process is not homogeneous anymore! This, however, makes complete sense. We have actually made the center of every realization special, meaning that homogeneity is broken in this sense. In fact, the new covariance is isotropic with respect to φ=0, further stating that the center of the GRF is somehow different from the rest of the points. All the presented machinery works not only for minima, but also for maxima and saddle points as well; the only difference among these being the sign of each λi. A.6.2 Distribution of heights and eigenvalues of the Hessian at a critical point In order to calculate the probability distribution of the eigenvalues of the Hessian at a certain height of the potential at critical points we should pay attention to two ingredients. The first one is the fact that the height and the second derivatives are correlated, so we need to calculate the multivariate covariance function for these quantities together. Furthermore, we also want to calculate this at critical points which can be done with the use of the generalized Kac-Rice formula. Assuming a critical point located at φ=0, the probability distribution to be computed is PV0, λ1, λ2∇V0=0(A.63) – 35 – JHEP05(2020)142 We can easily compute the PDF by conditioning the following random vector: {V0, h11, h22, h12, η1, η2}(A.64) of mean zero and covariance matrix    U2 0S02 0 S20 S22 0 0 0 S11   (A.65) Applying (A.19) and (A.20) to obtain the mean and covariance of the conditioned process and plugging them into (A.42), we get Pcp(V0, λ1, λ2)du 2 Y i=1 dλi=N |λ1||λ2|∆(λ)PV0, λ1, λ2∇V0=0(A.66) =N|λ1−λ2||λ1||λ2|exp −V2 0 2U2 0exp "−Λ2λi+V0 2U02#dλidV0 (A.67) where Nis a normalization factor and, in this two-dimensional example, ∆(λ)=|λ1−λ2|·π/2. Setting V0to a constant value, say V0=u, in (A.67) yields the distribution qu(λ1, λ2), defined in (A.42). On the other hand, integrating out either V0or the eigenvalues, gives the marginal distribution for the remaining variables in critical points (see appendix Bfor more detail). Another interesting application of (A.66) is that it can be used to count the expected number of critical points in a certain region of field space. For example, to compute the expected number of minima per correlation volume Λ2in the example above, a direct application of (A.37) yields E(#min) Λ2=Z+∞ −∞ du Z+∞ 0 dλ1Z+∞ 0 dλ2 π 2λ1λ2|λ1−λ2|PV0, λ1, λ2∇V0=0 =1 2√3.(A.68) In this case, the eigenvalues have been assumed to be positive. Setting other integration limits can give the expected number of maxima and saddle points, for example. A.7 Conditioned Gaussian random field for an inflection point We shall define an inflection point on our GRF as a point where the gradient of the field points in the direction of a Hessian eigenvector whose corresponding eigenvalue is zero. Furthermore, we will also demand that the non-zero eigenvalue of the Hessian to be positive at this point. In order to do this we can expand the discussion of the previous section by taking into account the third derivatives of the GRFs along with the lower ones. In order to simplify this description we will give a detail account of this construction for a 2dGRF – 36 – JHEP05(2020)142 only. Extending this to higher dimensions is straightforward. In particular we will be interested in the Gaussian random vector V(φ1), V (φ2), V0, V 0 1(0), V 0 2(0), V 00 11(0), V 00 22(0), V 00 12(0), V 000 111(0), V 000 122(0), V 000 222(0), V 000 112(0) (A.69) whose components have zero mean. As for the covariance matrix, it can be expressed as Σ =          U2 0C(φ1−φ2)C(φ1)SV1(φ1)SV2(φ1)SV3(φ1) C(φ2−φ1)U2 0C(φ2)SV1(φ2)SV2(φ2)SV3(φ2) C(φ1)C(φ2)U2 00S02 0 S1V(φ1)S1V(φ2)0S11 0S13 S2V(φ1)S2V(φ2)S20 0S22 0 S3V(φ1)S3V(φ2)0S31 0S33          (A.70) where (for the 2D case) SV3(φ) = −C0 111(φ)−C0 122(φ)−C0 222(φ)−C0 112(φ)=ST 3V(A.71) S13 = −α4−α22 0 0 0 0 −α4−α22 !=ST 31 (A.72) S33 =     α6α24 0 0 α24 α24 0 0 0 0 α6α24 0 0 α24 α24      (A.73) and the other matrix blocks have been defined in (A.46)–(A.52). Following the same steps as in the critical point case, we can obtain (for the covariance function (A.59)) the expression for a GRF once we conditioned everything up to the third derivative. In order to do this we can first compute the mean value of the GRF in the vicinity of our inflection point, which is given by ˜µ(φ) = 0+C(φ)SV1(φ)SV2(φ)SV3(φ)     U2 00S02 0 0S11 0S13 S20 0S22 0 0S31 0S33      −1     u η h ρ      (A.74) =C(φ)SV2(φ) U2 0S02 S20 S22 !−1 u h!+SV1(φ)SV3(φ) S11 S13 S31 S33 !−1 η ρ! = exp−φ2 2Λ2 (u+φ·η)1+ φ2 2Λ2+1 2 2 X i=1 λiφ2 i+1 6 2 X i,j,k=1 φiφjφkρijk ,(A.75) where the basis of φhas been chosen to be the eigenbasis of the Hessian matrix (whose components are described by hand its eigenvalues by λi) and we have denoted by ηand ρthe components of the first and third derivatives at the origin along the eigenbasis. – 37 – JHEP05(2020)142 The conditioned covariance, on the other hand, reads ˜ C(φ1,φ2) = C(φ1−φ2)−C(φ1)SV1(φ1)SV2(φ1)SV3(φ1)    U2 00S02 0 0S11 0S13 S20 0S22 0 0S31 0S33     −1    C(φ2) S1V(φ2) S2V(φ2) S3V(φ2)     =U2 0exp−|φ1|2+|φ2|2 2Λ2expφ1·φ2 Λ2−1−φ1·φ2 Λ2−(φ1·φ2)2 2Λ4−(φ1·φ2)3 6Λ6(A.76) which, once again, is isotropic around the origin of the field. A.7.1 Probability distribution for the inflection point parameters We can extend the treatment for the eigenvalues of the hessian that we did for the critical points to inflection points. The difference is that we will now impose that one of the eigenvalues vanishes while the other one is positive. Furthermore we will also impose that the gradient in the second eigenvalue direction also vanishes. These conditions have to be included in the calculation of the PDF of the parameters of the inflection points (V0, η1, λ2,ρ). Using a generalized version of the Kac-Rice procedure we arrive to, Pinf dV0dλ2dη1dρ=N|λ2|2|ρ111|PV0, λ2|λ1= 0 P(η1, ρijk |η2= 0) (A.77) where PV0, λ2|λ1= 0 dV0dλ2=Nexp −4V2 0−2Λ2V0λ2−Λ4λ2 2 2U0dV0dλ2(A.78) P(η1, ρijk |η2= 0) dη1dρijk = Nexp  −Λ2 12U2 0 18η2 1+ 6Λ2η1(ρ111 +ρ122)+Λ4 2 X i,j,k=1 ρ2 ijk  dη1dρijk (A.79) In (A.77), one of the |λ2|factors comes from the Jacobian of the variable change to the eigenbasis of the Hessian (though with λ1= 0); the remaining |λ2||ρ111|factor is just the determinant appearing in Kac-Rice’s expression. These last expressions can be used as in (A.68) to compute the expected number of inflection point per correlation volume Λ2, which yields, for our choice of covariance function, E(#ip) Λ2=√5−√3 3π.(A.80) B Numerical implementation and tests of the probability distributions B.1 Generation of Gaussian random fields: Karhunen-Lo`eve expansion In order to generate realizations of two-dimensional Gaussian random fields, we resorted to the so-called spectral or Karhunen-Lo`eve decomposition, due to its mathematical and computational simplicity. – 38 – JHEP05(2020)142 Given a certain mean function µ(t), covariance function C(t,s) and a discretized space {ta}(where aruns over all npoints in the lattice space) of a GRF, we can build the matrix Cab =C(ta,tb), which by construction is symmetric and positive definite; therefore, we can always decompose Cab as C=UΛUT(B.1) where Λ = diag(λ1, . . . , λn) is the diagonal eigenvalue matrix, consisting of non-negative entries, and Uis constructed by inserting all eigenvectors along its rows. Since Λ >0, we can further decompose Cas C=U√Λ√ΛUT=U√ΛU√ΛT=L LT.(B.2) This procedure is tantamount to performing a Cholesky decomposition [56] on C; which is by far the most expensive step in this algorithm, in terms of computational cost. Once we have computed L, constructing the GRF on the discretized space is straightforward. We only need to construct a random vector ξof length nwhose entries are independently distributed as Gaussian variables of zero mean and unit variance, and introduce the following variables: Va=µa+Labξb,(B.3) where µa=µ(ta). It can be easily shown that this gives the correct correlations among the values of the GRF evaluated at different points ta, h(Va−µa)(Vb−µb)i=hLacξcLbdξdi=LacLbdhξcξdi =LacLbdδcd =LacLbc =LacLT cb = (LLT)ab =Cab =C(ta,tb).(B.4) The main advantage of using this procedure to generate GRFs is that the main computationally costly step, constructing the Lmatrix, needs to be performed only once. The rest of the algorithm is highly trivial from this perspective and allows for further simplification, as we have seen. B.2 Numerical evaluations of critical points Using the expressions above we can compute the normalized distribution of heights of minima, maxima and saddle points for a 2dGRF, Pu,mindu =√3 4πU0e−u2/U2 0−2u U0+2√πeu2/4U2 0erfchu 2U0i+√2πu2 U2 0−1eu2/2U2 0erfchu √2U0idu Pu,maxdu =√3 4πU0e−u2/U2 02u U0+2√πeu2/4U2 0erfch−u 2U0i+√2πu2 U2 0−1eu2/2U2 0erfch−u √2U0idu Pu,sp du =√3 2√πU0exp−3u2 4U2 0.(B.5) – 39 –