Tensor modes on the string theory landscape

We attempt an estimate for the distribution of the tensor mode fraction $r$ over the landscape of vacua in string theory. The dynamics of eternal inflation and quantum tunneling lead to a kind of democracy on the landscape, providing no bias towards large-field or small-field inflation regardless of the class of measure. The tensor mode fraction then follows the number frequency distributions of inflationary mechanisms of string theory over the landscape. We show that an estimate of the relative number frequencies for small-field vs large-field inflation, while unattainable on the whole landscape, may be within reach as a regional answer for warped Calabi-Yau flux compactifications of type IIB string theory.

String theory is a candidate for a fundamental theory of nature, providing at the same time a UV-finite quantum theory of gravity and unification of all forces and fermionic matter.
Mathematical consistency requires string theory to live in a ten dimensional space-time, and a description of our large four-dimensional physics thus necessitates compactification of the additional six dimensions of space.
The need for compactification confronts us with two formidable consequences: Firstly, even given the known internal consistency constraints of string theory, there are unimaginably large numbers of 6d manifolds available for compactification. Secondly, many compact manifolds allow for continuous deformations of their size and shape while preserving their defining properties (such as topology, vanishing curvature, etc) -these are the moduli, massless scalar fields in 4d. This moduli problem is exacerbated if we wish to arrange for low-energy supersymmetry in string theory, as compactifications particularly suitable for this job -Calabi-Yau manifolds -tend to come with hundreds of complex structure and Kähler moduli.
Therefore, a very basic requirement for string theory to make contact with low-energy physics is moduli stabilization -the process of rendering the moduli fields very massive.
Moreover, as supersymmetry is very obviously broken -and so far has not been detectedideally, moduli stabilization should tolerate or even generate supersymmetry breaking. And finally, the process should produce a so-called meta-stable de Sitter (dS) vacuum with tiny positive cosmological constant, so as to accommodate the observational evidence for the accelerated expansion of our universe by dark energy [1,2,3].
The task of moduli stabilization and supersymmetry breaking has recently met with considerable progress, which is connected to the discovery of an enormous number [4,5,6,7,8] of stable and meta-stable 4d vacua in string theory. The advent of this 'landscape' [7] of isolated, moduli stabilizing minima marks considerable progress in the formidable task of constructing realistic 4d string vacua.
A large aspect of these recent advances relies on the use of quantized closed string background fluxes in a given string compactification. These flux compactifications can stabilize the dilaton and the complex structure moduli of type IIB string theory compactified on a Calabi-Yau orientifold supersymmetrically [5]. The remaining volume moduli are then fixed supersymmetrically by non-perturbative effects, e.g. gaugino condensation on stacks of D7branes [6]. The full effective action of such fluxed type IIB compactifications on Calabi-Yau orientifolds was derived in [9]. In type IIA string theory on a Calabi-Yau manifold all geometric moduli can be stabilized supersymmetrically by perturbative means using the larger set of fluxes available [10].
Moduli stabilization itself is a necessary prerequisite for a successful description of cosmological inflation in string theory. For slow-roll inflation requires a separation of scales between the inflationary scalar degrees of freedom with masses lighter than the inflationary Hubble scale and every other scalar field which needs to be heavy. Beyond that, the slow-roll flatness of the scalar potential required during the last observable 60 e-folds of inflation requires a substantial amount of control over higher-dimension operators in the effective field theory. This further motivates embedding inflation into string theory as a UV-complete candidate for quantum gravity.
Counting numbers of models, most successful inflationary model building in string theory has focused on the corner of the landscape described by type IIB flux compactifications on orientifolds of warped Calabi-Yau threefolds, and has produced small-field models of slowroll inflation. A small-field model is characterized by the fact that the field range ∆φ 60 the inflaton traverses during the observable last 60 e-folds of inflation is less than M P . Planckian field traversal during inflation marks a critical boundary, as here one transitions from the need to control chiefly dimension-six operators in small-field inflation to the need to control correction to all orders in large-field inflation. The inflaton candidate in these constructions is often chosen to be the position of a mobile D-brane [11,12], a combination of the geometric volume moduli of the Calabi-Yau [13,14,15,16,17], or an axion originating in the higher p-form N SN S and RR gauge fields of string theory [18,19,20,21].
There are several alternatives this plethora of slow-roll small-field models. One comes in the form of DBI inflation, where the specific form of the interactions of the inflaton dictated by the DBI action on a D-brane serve to slow down the field on a steep potential [22].
Another one consists of the idea of using a 'coherent' assistance effect of typically hundreds of string theory axions with sub-Planckian field range to yield an effective large-field model, called 'N-flation' [23]. Finally, there are recent constructions harnessing monodromy of the potential energy sourced by branes or fluxes with respect to D-brane position or p-form axions. This monodromy inflation mechanism allows for parametrically large-field inflation in string theory driven by a single field [24]. In the case of axion monodromy, the powerful shift symmetries of some of the p-form axions allow for a well-controlled and potentially large class of large-field models on warped type IIB Calabi-Yau compactifications [25,26,27,28].
For some recent reviews on flux compactifications and the associated questions of the landscape of meta-stable dS vacua and inflation in string theory, with a much more complete list of references, please see [29,30,31].
Observationally, the ∆φ 60 ∼ M P boundary between small-field and large-field models has a second tantalizing aspect. This is the case because of the Lyth bound [32,33] shows that for any single-field model of inflation, observable tensor modes in the CMB require a super-Planckian field range. Upcoming CMB observations, both ground and space based, are projected to detect or constrain the tensor to scalar ratio r at the level r 0.01 (see for example [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]), while existing and upcoming satellite experiments also significantly constrain the tilt of the spectrum [49,3,50].
The tensor-to-scalar ratio r is a single dimensionless number with a lot of discriminative power, separating single-field slow-roll inflation into two classes clearly distinguished by their dramatically different sensitivity to UV physics. Each of the two classes encompasses a large set of individually different models of inflation. Therefore, one may hope by summing over sizable samples of each class -each sample averaging widely over many different model construction corners of the landscape -and accounting for the influence of their respective population likelihood by cosmological dynamics, like tunneling, one can provide a statistical expectation as to whether r = 0 or r > 0 (in the observationally accessible sense of r 0.01). If successful, this would provide a second example where statistical reasoning on the landscape leads to a modest prediction akin to the weakly-anthropic explanation of the present-day small vacuum energy by scanning over N vac. 10 500 flux vacua [4].
The present work is an attempt to do so. We will show that an answer to this question can be reduced to the question of the number frequency distributions of small-field and largefield driving regions of the landscape, and of their vacuum energies -that is, counting. Then we will fail, as we do not know how to count across the whole landscape, for its largest part is terra incognita still. A much more modest version of the counting task can be formulated for the region of the landscape described by flux-stabilized warped CY 3-fold compactifications of type IIB string theory in the description of F-theory on elliptically fibered 4-folds. A large sample (several millions) of such potentially low-energy supersymmetric compactifications are fully computationally accessible in terms of hypersurfaces in toric ambient space described completely the discrete data of the associated gauged linear-sigma models (GLSMs). We then outline for this sector of the landscape how to formulate the counting problem, and discuss the implications of a possible answer. For this purpose, we start in Section 2 with an analysis of those premises of possible arguments which follow from what is currently known about scalar fields in compactified string theory. Section 3 proceeds from these premises with an argument which essentially states that populating large-field models of inflation in string theory would cost exponentially dearly compared to seeding small-field models if populated from the lowest-lying de Sitter (dS) vacuum of the whole landscape.
Section 4 reviews the interplay of the dynamics of eternal inflation and tunneling as well as the anthropic requirements for a statistical explanation of the observed small cosmological constant (c.c.) to work. Its central outcome is the observation that the progenitor vacua of cosmologically and anthropically viable descendant regions of the landscape are of high-scale vacuum energy, and ultimately lead to democracy on the landscape. Hence the reduction to counting. Section 5 then closes with a discussion of these arguments.

Assumptions
Let us being by collecting some of the known results on scalar fields obtained from compactification of string theory to four dimensions. These results and properties of the types of scalar fields, the moduli, coming from a given compactification will form the premises of our later discussion of the prevalence of small-field versus large-field models of inflation in string theory.

Need for symmetry
We will start our walk through the premises by looking at the the need for a symmetry if large-field inflation driven by a single scalar field is to be embedded into string theory.
Large-field slow-roll inflation driven by a single scalar field is stable under radiative corrections only in presence of an effective symmetry suppressing higher-dimension operators to all orders, which must be dominantly broken by just the inflaton potential itself. This is immediately clear from the known argument to the effect of the so-called 'eta problem'.
The classical background dynamics of single-field large-field inflation requires an inflaton potential dominated by a monomial Such a potential requires a minimum field range needed to generate the last observationally required N e 60 e-folds of inflation of The minimal distance in field space traversed during the observationally accessible period of inflation is super-Planckian, that is 'large-field', in these models.
In absence of any further information constraining the effective field theory below M P dimension-6 operators will be generated by quantum corrections. Among them we will generically operators of the type Such a correction will shift, in particular, the 2nd slow-roll parameter where () ≡ ∂/∂φ by a piece which destroys slow-roll inflation as soon as the inflaton moves about a Planck unit.
The only known way to forbid these dangerous higher-dimension operators to all orders in perturbation theory for an elementary scalar field (besides supersymmetry or conformal symmetry which, however, are incompatible with positive vacuum energy) is a shift symme- try. An unbroken shift symmetry requires a constant scalar potential. Assume now, that the dominant source of soft breaking of the shift symmetry is the field-dependence of the scalar potential itself, and its potential energy density is sufficiently below the cutoff of the effective field theory. Then radiative corrections induced by this inflationary soft shift symmetry breaking are proportional to powers of the order parameter of the symmetry breaking, (2.6) So they become large only at extremely large super-Planckian field values if V 0 (φ 60 ) M 4 P . This is the original idea of large-field chaotic inflation [51].
Parametrically large-field inflation with a single scalar field in string theory will thus constrain us to finding candidate scalar field protected by a very good shift symmetry. This essentially limits us to the axions of string theory, as we will see.

Properties of scalar fields in compactified string theory
Let us now look at the general properties of scalar fields arising from compactification of string theory to four dimensions. 1 These consist of the geometrical closed string moduli related to massless deformations of the internal compact manifold, open string moduli such as transverse positions of D-branes mutually supersymmetric and BPS with respect to the background manifold, and pseudoscalar 'axions' from the NSNS and RR p-form gauge fields.
The geometrical closed string moduli, such as volumes or shapes of the internal manifold, are not protected by fundamental shift symmetries. 'Fundamental' is used here in the sense, that it describes a shift symmetry of a pseudoscalar field which is inherited from a gauge symmetry already present on the worldsheet. Geometric moduli therefore are generically not useful for parametrically large-field behaviour, although they allow very well for a plethora of small-field string inflation models. 2 Open string moduli, i.e. the position of the D-branes, have been shown in general to possess a sub-Planckian field-range for the canonically normalized scalar field corresponding to a given D-brane position modulus. Canonical normalization of the open string moduli involves inverse powers of the size of internal manifold which for control reasons must be always larger than the string length √ α [52]. Therefore 1 We do neglect the dilaton here despite it being there already in 10d because by definition it couples to everything and cannot be a good large-field inflation candidate. 2 In some cases, the structure of string compactifications on Calabi-Yau allows for a so-called 'extended no-scale structure' of the Kahler potential of the volume moduli which at large volume leads to suppression of the otherwise large 1-loop string corrections. In such cases, this enables the use of the certain volume moduli to generate an inflaton potential of the type which resides at the borderline between small-field and large-field models (e.g. fibre inflation in type IIB [16] The remaining class of scalar fields from compactification are axionic fields which arises from integration of the NSNS 2-form gauge field B 2 or RR p-form gauge fields C p over cycles of the internal manifold. This gives rise to axion fields where Σ i 2 denotes i th 2-cycle, and Σ α p denotes the α th p-cycle. The gauge symmetries of the pform gauge fields from the worldsheet translate into shift symmetries of the dual pseudoscalar 4d axion fields. These continuous shift symmetries are broken to a discrete subgroup by instanton effects, and sometimes also by the effects of orientifold projections introduced to further break supersymmetry. An example of the latter is type IIB compactified on an O7 orientifolded Calabi-Yau manifold with fluxes and D3-branes, where the orientifolding breaks the shift symmetry of axions coming from B 2 [24,25,26,27,28]. The p-form induced axions in string theory, denoted collectively with a I , are therefore periodic on fundamental domain with limited field range [53,54,20] ∆a I = (2π) 2 .

Populating vacua -tunneling and quantum diffusion
The final premise we need to discuss concerns the mechanism to populate meta-stable a given dS vacuum, and by extension a string theoretic inflationary region, in the landscape. The notion of vacua as the (meta)stable ground states of a local QFT as an effective field theory derived from string theory exists only the regime of controlled 4d low energy approximation to string theory. This is realized only in the large-volume and weak string coupling regime, when the supergravity approximation supplemented by the leading string-loop, α -and nonperturbative corrections is valid. Within this region of controlled approximations the only known mechanism of vacuum transitions at zero temperature proceeds via field theoretic tunneling.
There are basically two known Euclidean instantons describing tunneling in QFT, and one process based on the quantum fluctuations of a light scalar field in dS space. The string landscape consists of a moduli scalar potential for an O(100 . . . 1000)-dimensional scalar field space which has upwards of O(10 500 ) isolated local minima. The local situation of tunneling between two adjacent vacua in moduli space is the often described by a scalar potential V (χ i ) of the canonically normalized moduli χ i which has two local minima χ i,± separated by a finite potential barrier. Let us call χ i,+ and χ i,− the false, and true vacuum, respectively.
These minima are separated by generically sub-Planckian distances in field space If in this general situation the barrier height is non-negligible compared the vacuum energy difference ∆V = V (χ i,+ )−V (χ i,− ), then the dominant Euclidean instanton contributing to tunneling is the Coleman-DeLuccia (CDL) instanton [55,56]. In flat space this instanton is described by the so-called SO(4) symmetric 'bounce solution' to the Euclidean field equations in the 'inverted' scalar potential −V (χ i ) Here ρ = τ 2 + | r| 2 denotes the SO(4) symmetric radial variable with τ = it Euclidean time. The boundary conditions on the bounce solution require In terms of ρ, χ i (ρ) they read as One then computes the Euclidean action on the bounce solution, called here χ (2.14) and the tunneling rate is given by The presence of gravity yields corrections to this result which become important in two situations. For one, this happens if the barrier thickness becomes super-Planckian, which is generically not the case for next-neighbour local minima in the landscape moduli potential.
The other situation occurs when the false vacuum approaches zero vacuum energy with the true vacuum being AdS. Then gravitational suppression of tunneling can happen. This case is irrelevant for our situation of having to populate a possible inflationary region of the landscape from a prior, higher-lying nearby dS vacuum. Therefore, CDL tunneling in our dS-to-dS situation with sub-Planckian barrier thickness can be described by flat space CDL tunneling neglecting gravity.
This generic picture of a landscape populated CDL tunneling has been studied extensively in the literature. In certain controlled constructions such as warped Calabi-Yau compactifications of type IIB string theory with imaginary self-dual 3-form fluxes [5] and anti-D3-branes provide a string theoretic realization of the effective CDL tunneling description in terms of the derived moduli potential [57,58].
The exception to this situation is the case where the barrier becomes very shallow and flat. In that case the gravitational corrections to tunneling become very important, and the Euclidean solution with the smallest action B mediating tunneling switches to the Hawking-Moss instanton [59]. This describes tunneling from the false vacuum at χ i,+ to the barrier top at χ i,T with a rate . (2.16) Finally, one can show that this process is essentially equivalent to a description where the light scalar fields χ i close to the false dS vacuum at χ i,+ are driven by the dS quantum fluctuations up the barrier onto its top. The magnitude of the dS quantum fluctuations which form a Gaussian random field are given by (see e.g. [60]) where H + = V (χ i,+ ) denotes the Hubble constant of the false dS vacuum. The diffusion probability to reach the barrier top behaves like the one derived from the Hawking-Moss instanton.
For all processes it is visible that a possible minimal field displacement will be expensive in terms of tunneling rate suppression if such a displacement due to tunneling were required by the initial conditions of a given inflation model.

An almost argument -field displacement is expensive
We will now explore the consequences of the premises outlined in the last section.
We know from the discussion in subsection 2.1 that large-field inflation driven by a fundamental scalar field needs an effective shift symmetry to be radiatively stable. The only other known mechanism for curing the radiative instability of generic scalar field theories, supersymmetry, forbids positive vacuum energy which renders it useless for inflationary purposes. From subsection 2.2 we have that essentially all scalar fields from compactifying string theory to four dimensions will have a sub-Planckian, or in some case just-so Planckian, intrinsic field range. Now consider that, in particular, the fundamental domain of all scalar fields with good shift symmetries from string theory, the p-form axions, is limited. Then by the very definition of a fundamental domain the only way beyond this point consists of finding an effect which unwraps the fundamental domain onto its covering space. This is called monodromy. For inflationary purposes this indicates two things. Firstly, the required effect has be something which gives potential energy to the candidate inflaton field in a given string compactification.
Secondly, it must possess monodromy in the inflaton in order to see its covering space instead of its limited fundamental domain.
This would result in parametrical large-field inflation (kinematically at least, up to back reaction constraints). Some form of non-trivial monodromy of the potential energy of the candidate inflaton with respect that very same inflaton is thus necessary for large-field inflation. We see, that this follows from the very definition of the limitation of the field range for those fields, the axions, which possess potentially good shift symmetries.
As parametrical large-field inflation in string theory can only proceed given a good shift symmetry, the only viable candidates are the p-form axions and we are left with trying to generate axion monodromy in some form of potential energy for the axion. A potential energy for the p-form axions -which spontaneously breaks their shift symmetry -is generated by instanton effects, branes or fluxes. Monodromy in the potential energy with respect to the p-form axions exists for (p + 3)-branes wrapped on p-cycles [25], or non-topological fluxes involving the p-form gauge field on a p-cycle [28]. The crucial aspect here is the fact that the monodromy-carrying objects, the branes and non-topological fluxes, spontaneously break the axionic shift symmetry the same way they also spontaneously break supersymmetry in a regime controlled typically by warping. Therefore, parametrical large-field inflation in string theory will come from some form of axion monodromy.
For the case of axions from c = Σ 2 C 2 in type IIB on a homologous pair of 2-cycles in a pair of warped throats wrapped by an N S5N S5-brane pair [25] this leads to a large-field The back reaction of the inflationary vacuum energy stored in the wound-up axion on the moduli potential as well as differing types of potential energy with axion monodromy such as other branes or fluxes will generically lead to a flattening-out of the axion inflaton potential [28]. In general we expect large-field potentials from axion monodromy to behave There is a crucial observation to be made here. The scalar potential during axion monodromy inflation is driving both the leading shift symmetry breaking and the back reaction.
Therefore, the point of vanishing axion vev being also the point of vanishing axion-induced D3-brane charge and potential energy is a minimum of the inflaton potential. This holds to high accuracy independently of the shape or structure of the moduli potential, or the back reaction of the compactification geometry. If it were otherwise, the non-universality of the back reaction from the moduli potential or the geometry would destroy the shift symmetry in a non-universal way to begin with. This, by construction, cannot happen as the sole source of back reaction is controlled by the same parametrically weak effect which spontaneously breaks the shift symmetry to leading order in the first place.
Let us denote the moduli sector other than the inflaton axion collectively with field(s) χ.
Then we can write the full scalar including both moduli and large-field inflation from string theory following from the premises of section 2 and the discussion above as The final aspect of our discussion below will involve the dynamics of populating a potential large-field inflation model in string theory. The only way known to exist proceeds via quantum tunneling from a prior meta-stable dS vacuum (see subsection 2.3). Thus, we will only need to discuss the immediate neighbourhood of the moduli potential with respect to our current vacuum. Tunneling will proceed dominantly from the closest-by higher-lying dS minimum of the moduli potential with the smallest barrier height at the point where the Euclidean bounce solution crosses the barrier. Again, this relies on the decoupling between the position of the minimum of the inflaton potential and the moduli potential due the strong axionic shift symmetry. Neglecting the generally curved trajectory of the multi-field bounce solution, we will take the moduli potential as approximated by a '1d section along the bounce'. Then the existence of two close-by non-degenerate dS minima can be modeled by a quartic polynomial Such a local neighbourhood structure of the moduli potential relies on two properties. Firstly, the string landscape admits an extremely large number of isolated minima of the moduli potential. Second, the intrinsic field range limitation of all moduli fields implies that most of these minima must have distance M P . This justifies Taylor expanding the potential around a given minimum towards the closest neighbour.
At last, there are subleading sources of shift symmetry breaking, typically non-perturbative effects. The presence of these instanton effects generates a periodic potential for the p-form axions. Its period is given by the extent of their fundamental domain, 2πf in terms of the axion decay constant f . The magnitude of the instanton-induced axion potential is exponentially suppressed at large volumes [61,25]. This gives us where vol Σ denotes the volume of the cycle threaded by the axion. We can thus easily dial them negligibly small. Yet their presence will be crucial for the relative count of small-field inflation models in string theory compared to the axion monodromy based large-field models.
The full scalar potential for large-field inflation in string theory in presence of the local moduli potential, under the premises of section 2, thus reads We will consider for the sake of explicitness V 0 (φ) of C 2 axion monodromy from N S5-branes, but the conclusions drawn from here will be general and do not depend on the precise choice This form introduces the slope parameter True large-field behaviour requires b 1, or equivalently vol Σ 2 1. For this case, the example potential is shown in Fig. 1. Let us label again the two adjacent (meta)stable minima of the moduli potential by χ ± with χ + < χ − and place χ − = 0.
Conversely, fine-tuning b around b 1 will have near-flat inflection points appearing in the potential [26]. As f < M P , these inflection will be spaced with sub-Planckian distances.
Therefore, moderately fine-tuning one of them to slow-roll flatness around the inflection point by using b will result in small-field inflation.
This leads to a crucial result: Under the premises of section 2, there will be at least one model of small-field inflation contained in every working model of large-field inflation in string theory.
The small-field and large-field parameter regions of axion monodromy occupy different volumes of microscopic parameter space, as the occurrence of a slow-roll flat inflection point needs a significant tuning in b 1 in terms of the microscopic parameters, such as fluxes. Such fine-tuning can range from moderate O(10 −2 ) [18,12] to more severe values of O(10 −8 ) [15]. However, the number frequency hierarchy deriving from the tune is finite, and can be easily dominated by exponential ratios from the dynamics of populating all these different models, which is tunneling.
We will now take a look at the process of tunneling into the inflationary valley of χ − = 0 from the close-by valley at χ + < 0. Behind this is the premise of subsection 2.3 that tunneling is the only process for cold vacuum transitions in the landscape which is known. within the quadratic region φ M P close to the origin of V 0 (φ). Unless the lowest-lying, and thus closest-to-origin inflection point is tuned inflationary flat using b 1, then there will be local minima at lower-lying inflection points which would trap the inflaton in false vacua. Conversely, the higher-lying inflection points above the fine-tuned one are too steep to support small-field inflation. An example of such an instanton-induced small-field contained in every stringy large-field model is shown in Fig. 2. The inflationary inflection points are the two ones closest to φ = 0.
We will first look at tunneling mediated by the CDL instanton in the flat space approx- imation [55,56]. As discussed in subsection 2.3, the gravitational correction factor from including gravity is typically not important for tunneling from one dS to another lower dS vacuum.
Let us first discuss which locale in the false-vacuum valley close to φ + = 0 we expect to be the most likely starting position for any tunneling process. We may a priori expect the false valley to get populated by an even earlier tunneling event. That event may exit in particular at some φ = 0 up the valley from where a further tunneling could start. However, the false valley is slow-roll in φ even more than the true valley close to χ − = 0, and it supports false-vacuum eternal inflation at its false minimum χ + < 0, φ + = 0. Therefore, the ambient space-time residing initially in the false valley will have a fraction exponentially close to unity which actually sits at the false vacuum. The false vacuum inside the false valley thus exponentially dominates the initial state for any subsequent tunneling towards the true valley.
Next, getting slow-roll inflation in the true valley after tunneling there places a constraint on the exit state directly after tunneling. We must have the position of φ after tunneling, called φ 0 , supporting at least 60 e-folds of slow-roll inflation in the true valley For the large-field case b 1 this implies while in the small-field situation with b 1 we have Note, that the fields may exit at χ 0 < 0 and |φ 0 | φ 60 and possess finite speeds φ (ρ), χ (ρ) as well, even in the small-field case. This is due the the negative spatial curvature inside a freshly formed CDL bubble. Negative curvature will serve to slow scalar fields regardless of their initial conditions enough to track them into slow-roll even on a small-field inflection point [62,63] 3 . A viable bounce thus requires boundary conditions according to eq. (2.13) and for reasons of regularity also There is also an energetics constraint which reads to get down tunneling.
If we look now at the inverted potential −V (φ, χ) driving the Euclidean dynamics, eq. (2.11), we see that such a bounce which originates downhill from the ridge and ends uphill on the ridge at χ + < 0, φ + = 0 does not exist. The gradient of −V (φ, χ) always points away from φ = 0. Even allowing finite initial speed φ (0), χ (0), neglecting the regularity boundary conditions eq. (3.29), does not avoid this fate, as the friction term in the bounce e.o.m. eq. (2.11) immediately destroys any initial speed [63].
This forces us to consider processes which move the field φ uphill in the false valley to values |φ| > φ 0 > φ 60 . A CDL bounce ending here and thus curving downhill (in the inverted potential −V (φ, χ)) from its starting point φ 0 , χ 0 is then possible after φ has moved uphill (in V (φ, χ)) in the false valley.
How do we move up the false valley? The false vacuum at χ + < 0, φ + = 0 drives falsevacuum eternal inflation. As m φ < H, φ undergoes scale-invariant dS quantum fluctuations.
These can be thought of as φ performing a Gaussian random walk with variance eq. (2.17) This quantum diffusion has small probability for large jumps ∆φ given in eq. (2.18) As m φ 10 −5 M P from the inflationary constraints in the true valley, we see that Therefore we arrive at a combination of two results. For every large-field inflation model in string theory there exists at least one small-field model contained within via moderate tuning. However, the relative probability of realizing them dynamically via first quantum diffusing uphill in the false valley and then CDL tunneling out, is given by and typically N e 60 observationally.
We can check this result by replacing the quasi-rectangular path just considered by looking at quantum diffusion directly from the false vacuum at (φ + , χ + ) to a point (|φ| > φ 60 , χ T on the top of the potential barrier ridge at χ T (which is then followed by classical rolling into the true valley). This process, as discussed in subsection 2.3, is equivalent to a Hawking-Moss instanton [59] which tunnels to said point on the barrier ridge. Both descriptions yield parametrically the same result as found above in eq. (3.34).
This exponential advantage of the instanton-induced small-field regime crucially depends is constant over moduli space to a high degree However, its magnitude can and will generically vary widely over moduli space. This happens because changing b will preserve the positions of the critical points of the instanton correction which is what the discrete remainder of the shift symmetry demands An example of a potential from the axion monodromy large-field mechanism of string theory, which represents this generic situation on the landscape is depicted in Fig. 3.
This changes the picture in one crucial aspect. Tunneling in moduli space can now easily provide for access to the true-vacuum valley of slow-roll inflation, where b 1, from a falsevacuum valley with b > 1, as b will generically change upon barrier traversal. Therefore, in the moduli potential barrier, as they do not require uphill tunneling for their respective population.
We end up in the following situation. In all regions of the landscape where b varies freely over moduli space between b < 1 and b > 1, the small-field and large-field regime contained as one-to-one in all large-field models of string theory are populated equally by tunneling.
Therefore, the large-field mechanism by itself does not predict any bias towards small-field or large-field inflation, respectively.
The exception to this statement are possibly special regions in the landscape where microscopic constraints may impose b < 1 for, say, certain classes of compactifications.
There is no small-field inflation at all in the large-field models in such a landscape region.
However, the population of all possible large-field models in this region must proceed via uphill tunneling from φ + = 0. We can then apply the same type of calculation as before to axion monodromy large-field models with varying monomial power, and suppressed instanton corrections. In these special landscape regions the above result implies a hierarchy among the large-field models in string theory. This hierarchy exponentially favors the models with the smallest monomial power p The question whether small-field or large-field inflation is dominant in the landscape therefore has no answer within the large-field mechanism of string theory itself. The situation is better only for those presumably small regions of the landscape, which are characterized by bounded instanton corrections such that among them the large-field mechanism does not contain a small-field regime. We are thus forced to look to the far wider class of small-field models outside the range of the large-field mechanism, and we need to count -and eventually weight that count by the combined dynamics of tunneling and eternal inflation.
We may now ask about conceivable loopholes in the line of thought of this section. One known alternative to extending the field range seen by the potential energy of an otherwise periodic single axion, by definition, via monodromy, is an assistance effect of many sub-Planckian axions, known as 'N-flation' [23]. For this proposal to succeed, however, all of the axions used need a shift symmetry of similar quality as in single-field monodromy. The minimum of the combined effective multi-axion potential thus is similarly decoupled from vacuum-hopping in the moduli potential as in the single-field case -again requiring super-Planckian uphill quantum diffusion in the axion valleys to enable entering the inflationary multi-axion valley via tunneling. This results in the same exponential bias against N-flation large-field models, as for single field axion monodromy setups.
Next, it is clear that the minimum of the inflaton potential in large-field models is almost decoupled from the moduli potential by virtue of the shift symmetry, but not completely so.
Assume therefore that the minimum of the inflaton potential shifts by δφ M P each time a next-neighbour vacuum transition executes in the moduli potential. Then there will possibly be multi-tunneling paths through the landscape which transport us into a progenitor falsevacuum axion valley with a minimum of the axion potential φ min. φ 60 compared to the minimum of the final 'our-world' axion valley with φ min. = 0. A tunneling jump from such a progenitor valley will not require φ quantum diffusing up the hill by a super-Planckian field range, and would thus cause no relative suppression compared to the instanton-tuned small-field sub-setup. However, as the shift of the axion minimum δφ M P is very small, we will need many such tunneling jumps to get us into the right progenitor false-vacuum axion valley. As the maximum total potential difference ∆V tot crossed by N such jumps prior to arriving in the right progenitor valley in the controlled region of the landscape is ∆V tot. < M 4 P we have an average potential energy difference per jump of (3.41) In the limit of large N we get ∆V N → 0. Then the thin-wall limits holds for the N successive CDL bounces which implies As the full amplitude will be this will be exponentially suppressed for a super-Planckian ∆φ 60 M P compared to a smallfield model requiring ∆φ 60 M P , as then N large−f ield N small−f ield 1. This last hierarchy holds because the shift of the axion minimum due to a single-jump in the moduli potential is δφ M P . We have also assumed here that each of the N jumps traverses on average the same typically sub-Planckian field interval in moduli space, with the interval length not or only weakly depending on N . This seems to be reasonable as long as the number of jumps N is small compared to number N vac. 10 500 the landscape must possess in order to allow for a weakly-anthropic explanation of the present-day small vacuum energy.
We have seen in the last section that within the largest part of the landscape the mechanism for large-field inflation in string theory provides for large-field and small-field inflation regimes with comparable likelihoods because they get populated evenly by CDL tunneling. We will see that such a democracy between large-field and small-field inflation seems to hold generally across the landscape as far as the dynamics of eternal inflation and tunneling are concernedas long as we post-condition our analysis on the regions of parametrically small-c.c. dS vacua and inflationary regions which are cosmologically viable and permit an anthropic explanation of the observed extremely small c.c. along Weinberg's argument. We start with discussing the effects of tunneling and eternal inflation together with anthropic post-conditioning, and then proceed to discuss attempts at counting the different inflationary realizations on the landscape.

Eternity
There a few salient facts concerning the dynamics of tunneling and false vacuum eternal inflation which seem to hold across the whole landscape: • I) All inflationary regions in the landscape will eventually get seeded via tunneling.
The meta-stable dS vacua will undergo false-vacuum eternal inflation. This situation could only reverse itself if large-c.c. dS vacua are more than exponentially rare, or down-hill tunneling from large-c.c. dS vacua were forbidden.
Property III prevents this from happening. Furthermore, the characteristics III and IV combined tell us that the immediate field space neighbourhood of any small-c.c. dS vacuum will almost everywhere consist of large potential barriers and adjacent large-c.c. dS vacua.
Therefore, population of the small-c.c. dS vacua will happen almost everywhere by direct and (compared to up transitions from extremely small-c.c. vacua) fast down-hill tunneling transitions from adjacent large-c.c. dS vacua. Combined with I and II we also see, that such fast down-hill tunneling is necessary in order to have a shot at efficiently populating an exponentially large number of small-c.c. dS vacua for Weinberg's argument to work.
We can now compare this with the dynamics of eternal population. Appendix B concerns itself exclusively with the one relevant aspect here -the typical vacuum energy of the progenitor dS vacua. The discussion of this aspect follows in particular [67]. The upshot can be summarized as this: For global measures with full volume weighting the progenitor is the largest-c.c. metastable dS vacua. This is the result of the exponential 3-volume reward driving the progenitor Hubble parameter to be as large as possible. All other measures free of obvious paradoxa (global measures without exponential volume reward, such as the scale factor measure, or local measures, such as the causal diamond measure) will see the the landscape populated from the longest-lived progenitor dS vacuum.
From the discussion above, and in Appendix A we know that parametrical longevity of dS vacua is achieved once it has no down-hill tunneling paths accessible, and an exit thus will proceed by an up transition. Therefore, the longest-lived progenitor of a given region in the landscape will be a dS vacuum of somewhat small c.  c.c. V − . According to [6] we can write We tunnel into different descendant small c.c. dS vacua whose vacuum energies V − , V − differ by a still parametrically small amount |∆V − | V + . Hence, we immediately get that the negative term in B gets only a negligible correction from the variation of the vacuum energy among the many small-c.c. descendant vacua. The Euclidean action for the bounce solution itself we will approximate from above, and find the same parametrical result. We have roughly where ∆ξ is the Euclidean time elapsed while ρ(ξ) increases from zero to ρ max and decreases back to zero. From the CdL Einstein equation we get ρ max at dρ/dξ = 0 to be Moreover, we have ∆ξ H −1 + = 3/V + . Hence, we arrive at with the same parametrically suppressed correction. One can further show using the field e.o.m. that also the bounce solution φ(ξ) gets parametrically small corrections of O(∆V − /V + ).
In the thin-wall limit where we have not just we can calculate the influence of a shift |∆V − | V + in V − on B exactly and get in agreement with the general result above.
Summing up, we know that the anthropically efficient population of a cosmologically viable small-c.c. region of the landscape proceeds via tunneling from a very high-scale c.c. dS vacuum, and the down-hill tunneling rate into such small-c.c. vacua is independent from the varying vacuum energy of the many small-c.c. vacua. This implies that the tunneling rates get their variation from the distribution of barrier shapes -with the two main parameters height V T and thickness ∆φ -across the many down-hill tunneling paths from the high-scale progenitor to the exponentially many small-scale c.c. descendants.
Since the rates are independent from the varying c. Let us compare this to the discussion in section 3. From there we know that the smallfield regime inside stringy large-field models has to use the lowest lying instanton-induced inflection point, because doing otherwise would get the inflaton trapped in local minima at lower-lying inflection points. However, recently the absence of the overshoot problem was shown for tunneling-born small-field models [63]. This implies that we can tunnel into the small-field regime of a large-field model as far out and high (in potential energy) above the inflection point as we wish to. We can therefore, using the notation of section 3, seed both a small-field regime within a large-field model of string theory, as well as its own large-field regime by tunneling from an instanton-induced local minimum close to φ + ∆φ large−f ield As presented, this argument still has a possible loophole. Everything we said holds strictly true if the lowest-lying 1st inflection point is the only one suitable for small-field inflation. However, the latter statement is only valid for convex large-field potentials V 0 (φ) ∼ φ p , p ≥ 1. For concave potentials V 0 (φ) ∼ φ p , 0 < p < 1, however, which are still largefield for p 0.1, you could equally well have small-field on the n th inflection point at φ n th inf lection point > M P . This follows because almost all (except for the first few ones at φ M P where V 0 (φ) ∼ φ 2 again) of the lower-lying inflection points will have b < 1 as V 0 (φ) is concave. The number of inflection points generated by the instanton correction within ∆φ 60,large−f ield is given by We have already discussed that f M P in string theory, and in concrete models of 5-brane axion monodromy one gets e.g. [26] Here V is the warped volume of the internal manifold in units of α . Thus, N inf lection point 1 typically, and for concave models each of them is available for slow-roll tuning. This might lead us to conclude towards a counting bias towards small-field models. However, the same fact N inf lection point 1 also implies that the inflection points are spaced densely compared to the evolution of V 0 (φ). Therefore, if the n th , n 1 inflection point is tuned flat (b = 1), than its neighbours will have b 1 to very good degree, too. This leads to a wide field range over which the slow-roll parameters , η have sizable oscillations. They, in turn, imprint themselves as large oscillations on the 2-point function power spectrum of the curvature perturbation generated, which are severely bounded by the observed the CMB [26]. In particular, these limits imply b 1 [26], and therefore small-field inflation starting from a high-lying b = 1-tuned inflection point would give a universe with the wrong CMB. This removes all such small-field candidates except the ones starting from the few lowest-lying inflection points from the comparison with the large-field regime. So, demanding consistency with the observed CMB leads us back to the conclusion already drawn above.

Vacuum energy distribution
This democratic result is to be compared with the product of the number frequency dis-   (for a review, see e.g. [29]). We have M 2 S ∼ F in terms of the supersymmetry breaking F-terms, and the upper limit V max of the positive vacuum energy of a given vacuum is related to the F-terms as V max ∼ F 2 . Hence, we can estimate the largescale distribution of vacuum energies as the one given for M 2 S provided that no strong tuning of the vacuum energy has been selected for (the situation relevant for inflation). According to the results of [69,70,72], this leads to In flux compactifications we expect F ∼ W to have a flat distribution. This implies a flat number density distribution for the vacuum energy.

Multitude
We are thus left with estimating the number frequency distributions of small-field and largefield inflation mechanisms in the string landscape. Let us start with the generic small-field models on the landscape (i.e. those which do not arise from the instanton contributions to large-fields models with axion monodromy). Most of these occur 'accidentally', that is, in vacua where the microscopic parameters such as fluxes, result in a local inflationary slow-roll flat dS saddle point of the moduli potential which can drive inflation. Barring further constraints, we can as a very rough approximation model the landscape (outside the symmetry-protected large-field mechanism occurrences) as a random potential for an N -dimensional scalar field space. 7 We then need to determine how many slow-roll flat dS saddle points we statistically expect in such a description. This question has been dealt with in a work by Aazami & Easther [73], where the propose to model the landscape as a random potential given by while we get only Almost all of the critical points are saddles. For inflationary purposes we may wish to restrict our attention to class of saddles with just one negative eigenvalue, as these guarantee singlesmall-field inflation. For small cross couplings their number is There is by now ample evidence that the landscape contains, even in the small calculable . (4.63) One can then show [73] (see also more recently [76,77]) that the probability to have a local minimum (i.e. all positive eigenvalues of the Hessian) or a single-field saddle point is given As shown in [73], already a separation of scales between the f i and the cross couplings as small as 2 orders of magnitude is enough to sit safely within the first case discussed above, giving exponentially many local minima and single-field saddles potentially suitable for inflation.
In general, one expects these two cases to appear mixed together in that a few cross couplings may appear with M 10 −2 M P while most of them will be at smaller scales. The corresponding Hessian will then be approximately band diagonal, but the count of single-field saddle points will remain exponentially large of O(10 N ), because band width is generically expected to be small compared to N .
One may condition this analysis on subsectors of the landscape which potentially allow for low-energy space-time supersymmetry. In such a situation the random potential over moduli The upshot is that we will get in a landscape with N scalar degrees of freedom typically O(10 N ) meta-stable dS minima. The fraction β saddle of them which constitute single-field saddle points potentially suitable for inflation we do not know so far. From the existing random matrix studies [76] so far it is not clear whether there will be more or less single-field saddle points than meta-stable dS minima. What we do know about is the cost of flatness of such a saddle point. The fraction of them which are locally flat enough to support 60+ e-folds of slow-roll inflation will be determined by the fraction of volume in microscopic parameter space, such as fluxes, which yields sufficiently flat saddle points. According to work done by [18,12,15,75]  while the fraction of meta-stable minima is (4.68) In terms of these we can now estimate the number of local minima N i, min. on a given CY and the fraction of those which are sufficiently fine-tuned inflationary single-field saddle Here β i,f lat saddle denotes the ratio of the number of inflationary flat single-field saddle points to the number of meta-stable dS minima. Note that we do not know a priori whether β i,f lat saddle < 1 or β i,f lat saddle > 1. A more detailed study of random matrix models along the lines of [76] may yield an answer to this question.
To determine the fraction of manifolds with axion monodromy inflation we have to multiply each term eq. (4.69) with a factor δ which is either zero or unity depending on whether the given CY 3-fold has a projected-in RR C 2 -form axion (or equivalently, h 1,1 i,− = 1) and suitable source of potential energy with axion monodromy, everything placed inside a warped throat etc. For a conservative estimate we may ask to bound the number of CY's supporting axion monodromy by counting all those with h 1,1 i,− ≥ 1 as the most basic requirement (δ = δ h 1,1 i,− ≥1 ∈ 0, 1). We still do not know how to do this for all CY 3-folds. But, we may be able to do this for a large set (several million The sums in these expressions run over the set of CY's denoted by toric F − theory CY 4 s.
This result assumes a flat number frequency distribution of vacuum energy. Arguments for this flat prior to arise in the context of type IIB flux compactification were reviewed above in section 4.3. 9 Note that we do not know a priori whether β f lat saddle < 1 or β f lat saddle > 1. An estimate of the axionic in-projection cost β h 1,1 − ≥1 seems feasible for the large sample of CY 3-folds described in F-theory as elliptic 4-folds given in terms of their GLSM data. It is conceivable that a study of random matrix models along the lines of [76] may yield in fact β f lat saddle > 1.
Then in virtue of β h 1,1 − ≥1 bounding the number frequency of axion monodromy inflation from above, this would tell us to expect a negligible tensor fraction r in the type IIB CY landscape to the extent that the occurrence of an observable tensor-to-scalar ratio r 0.01 is tied to the inflationary scale and thus to the existence and realization of large-field models of inflation. 10 9 See, however, recent work on type IIA perturbative moduli stabilization [77], and type IIB Kahler uplifting [79], where a product probability distribution over several individually flat microscopic variables seems to give a vacuum energy distribution peaked around zero. Such a non-flat would have to factored into the result above eq. (4.73), giving a bias towards small-field models with their smaller vacuum energies. 10 The link between r 0.01 and large-field inflation is not watertight. On the one hand axion inflation can lead to additional highly non-Gaussian scalar perturbations sourced through the axion-photon coupling, which effectively suppresses r even for large-field models [80,81]. Next, small-field inflation models can be (severely!) fine-tuned to produce r 0.01 [82,83]. And finally, a small-field inflaton can source additional scale-invariant B-mode power through couplings to degrees of freedom (particles or strings) which get light at points of enhanced symmetry [84], similar to the trapping mechanism [85,86].
Let us stop here to summarize again the crucial aspects of the story just told. Firstly, the premises laid out in subsections 2.1, and 2.2 together imply that the properties of the scalar field from string compactification require large-field inflation in string theory to take the form of axion monodromy. The axionic shift symmetry, rooted in the p-form gauge symmetry on the worldsheet, decouples the position of the minimum of the axion monodromy inflaton potential from the moduli potential. If this were otherwise, it would imply sizable nongravitational couplings between the inflaton axion and the moduli which would invalidate the shift symmetry in the first place. Therefore, the many local minima of the moduli potential landscape share (almost) the same minimum of the axion inflaton potential.
Secondly, the population of the inflationary axion valley is only known to proceed within This counting is hard in general due to lack of calculational access. However, if we restrict the scope to first obtaining an answer for a region of the landscape with established control, counting may be feasible. As an example we gave a sketch of the discussion for the landscape of elliptically fibered 4-folds in F-theory. A large sample (several millions) of such potentially low-energy supersymmetric compactifications are fully computationally accessible in terms of hypersurfaces in toric ambient space described completely by the discrete data of the associated GLSMs. Hence, in this sector of the landscape we may be able to get an estimate of the fraction β h 1,1 − ≥1 of CY manifolds in the sample which have the RR-form axions required for axion monodromy large-field inflation in the first place (which we leave for future work).
As such, an estimate of β h 1,1 − ≥1 < 1 would provide an upper bound on the fraction of CY's which carry axion monodromy inflation. It is conceivable that a study of random matrix models along the lines of [76] may yield in fact that small-field models are more abundant than dS minima themselves. Combined with β h 1,1 − ≥1 < 1 this would imply the absence of detectable tensor modes if a detection of low-energy supersymmetry pointed towards CY's.
Finally, if there is a way of shifting around the axion minimum as a function of the moduli without spoiling the shift symmetry, or if there is a mechanism to protect large-field models without relying on an effective shift symmetry, the argument as it is fails.
Here we shall shortly discuss the process of tunneling uphill from a lower-lying dS vacuum into a higher-lying one. This process is highly exponentially suppressed compared to downhill tunneling, and as a function of the vacuum energy of the final higher-lying minimum. One can see this explicitly in three different regimes of CdL tunneling. As we have the hierarchy denotes the tension of the CdL bubble wall, with χ − denoting the position of lowest-lying dS minimum V − = V (χ − ). The ratio T 2 /V + controls the importance of the gravitational correction inside the rectangular bracket. If we approximate the potential barrier separating χ ± as being of height V T V + and thickness ∆χ we can write T ∼ ∆χ √ V T . Gravity is important for V + T 2 , or equivalently ∆χ For sub-Planckian barrier thickness we expect that the leading terms in the scalar potential which are responsible for the two adjacent local minima at χ ± will also produce the barrier separating them. Therefore, if V + is not subject to specific tuning, we expect the barrier height to vary roughly together with V + as V + V T . This leads to S strong grav.
with some c > 0. This is a regime where a hierarchy S strong grav.
is valid. If the barrier thickness ∆χ is sufficiently sub-Planckian, increasing V + while keeping V T fixed will eventually take us into opposite regime ∆χ V + /V T of weak gravity where S weak grav.
Thus, we see that increasing V + eventually leads to a regime where eq. (A.78) is again valid.
In summary, CdL tunneling in the thin-wall approximation yields a hierarchy leading to an exponential suppression of uphill tunneling scaling as Γ /Γ ∼ exp(−c/V + ) for V + < V + .
The exception is a situation where V + > 0 is tuned to be extremely small compared to the barrier height V T . However, this limit is irrelevant for the discussion here, as the discrimination between large-field and small-field inflation around ∆φ 60 M P corresponds to a change of the inflationary potential energy by about 2 orders of magnitude around the GUT scale.

A.2 CdL tunneling away from the thin-wall approximation
We saw in the last section how lifting V + towards V T shuts down the gravitational correction in the thin-wall limit. However, eventually this limit will also leave the thin-wall approximation itself. There are no general explicit results for the bounce action away from the thin-wall approximation known for generic potentials. However, one may approximation any given smooth potential with two local minima by triangulating it with linear functions.
Coleman tunneling in such an approximative piecewise linear potential can be solved exactly by analytical methods without using the thin-wall approximation [87]. In the case where ∆V + < ∆V − /4, the bounce action can be found to be [87] S E (χ) = 32π Here it is  those which do not immediately run into paradoxa such as the youngness, or the Boltzmann brain problems. However, for the purpose of our discussion their most important property is that they separate into two classes with respect to the one relevant aspect here -the typical vacuum energy of the progenitor dS vacua. The discussion of this aspect follows in particular [67].
• (G) One class (the so-called global measures) rewards different inflationary vacua or regions of the landscape proportional to the volume of 3-space generated during the eternal phase. Correspondingly, in volume-weighted global measures all the vacua and slow-roll inflationary regions of the landscape are seeded ultimately by the highest-lying meta-stable dS vacuum of the landscape.
Let us illustrate this at the example of a simple toy landscape with 3 vacua V s < 0 < V 1 < V 2 which is depicted in Figure 5. This example features the general structure of the string landscape -neighbouring vacua tend to have large differences in vacuum energy, the width of the potential barriers are M P , and there are AdS vacua.
Therefore, we choose the vacuum S to be an AdS vacuum of negative cosmological V Φ 2 1 S Figure 5: A simple toy landscape with 3 vacua V s < 0 < V 1 < V 2 . V s is an AdS vacuum and acts as a so-called 'sink', i.e. a vacuum where eternal inflation ends in a crunch.
constant V s < 0 . Tunneling from vacua with positive vacuum energy, such as the vacua 1 and 2 into vacuum S will create AdS bubbles within which space-time ends in a big brunch. Therefore, the AdS vacuum acts as a sink, destroying probability current flowing from the eternal inflating vacua 1 and 2.
The vacuum population dynamics of this system is governed by differential rate equations. They determine the rate of change of probabilityṖ i of realizing vacuum i by the probability currents J ij which feed or drain vacuum i [67,90] P 1 = −J 1s − J 12 + J 21 + J 1,vol (B.85) The probability currents are given as J ij = P i Γ ij , and J i,vol = P i · 3H i . Here, Γ ij denotes the decay rate for forming bubbles of vacuum j in a sea of vacuum i. Note, that in a global measure the volume growth ∼ e 3H i t of each vacuum i is weighted for by adding J i,vol . The dS-dS vacuum decay rates are given from CdL tunneling as while we denote the decay of vacuum 1 into the AdS vacuum S by Γ 1s = e −C 1 . From now on, we will set Γ 2s = 0 for simplicity. Then the vacuum dynamics readṡ P 1 = −P 1 (Γ 1s + Γ 12 ) + P 2 Γ 21 + 3H 1 P 1 (B.87) P 2 = −P 2 Γ 21 + P 1 Γ 12 + 3H 2 P 2 .
Furthermore, in most cases we have overwhelmingly H i Γ ij .
With these inputs, eq.s (B.87) has a solution [67] All vacua grow with the volume growth of the highest-lying meta-stable dS vacuum whose population dominates everything else. This does not depend on the decay rate Γ 1s into the AdS sink, as long as H 2 > Γ 1s .
• (L) Conversely, the other class (the local measures) discards rewarding the 3-space volume generated. They just account for the bare-bones anthropically required 60-odd e-folds of slow-roll volume growth. Local measures seed all vacua from the longest-lived progenitor.
Note that the volume growth rate contributions are absent by definition of the local nature of this class of measures. There is one variation to this class of measures, in that there is a local-global duality which links the local causal patch measures with the global 'scale-factor' measure. The scale-factor measure adds back volume growth terms J i,vol = 3P iṖ 1 = −P 1 (Γ 1s + Γ 12 ) + P 2 Γ 21 + 3P 1 (B.90) However, these lead to universal volume growth ∼ e 3t of all dS vacua. This implies, that the overall volume growth can factored out unambiguously, so that ratio of vacuum population probabilities behave exactly as in local causal patch measures (this is a manifestation of the 'global-local duality' between causal patch measures and the scalefactor measure [101]).
The asymptotic behaviour of the ratio P 2 /P 1 does now have two distinct regimes, depending on whether Γ 1s Γ 21 (a 'narrow' sink) or the opposite Γ 1s Γ 21 (a 'wide' sink) is realized [67,90]. For a narrow sink we find The opposite case of a wide sink yields Both cases share a common property -for a narrow sink, the vacuum populations are dominated by vacuum 1, while for a wide sink vacuum 2 dominates -in each cases it is the longest-lived dS vacuum which dominates the landscape in the stationary limit, and in turn then feeds everyone else [67,90].