N and the stochastic conveyor belt of Ultra Slow-Roll

∆N and the stochastic conveyor belt of Ultra Slow-Roll

Tomislav ProkopecInstitute for Theoretical Physics, Spinoza Institute and the Center for Extreme Matter and Emergent Phenomena ,

Utrecht University, Buys Ballot Building, Princetonplein 5, 3584 CC Utrecht, the Netherlands

Gerasimos RigopoulosSchool of Mathematics, Statistics and Physics, Herschel Building,

Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

We analyse field fluctuations during an Ultra Slow-Roll phase in the stochastic picture of inflationand the resulting non-Gaussian curvature perturbation, fully including the gravitational backreac-tion of the field’s velocity. By working to leading order in a gradient expansion, we first demonstratethat consistency with the momentum constraint of General Relativity prevents the field velocity fromhaving a stochastic source, reflecting the existence of a single scalar dynamical degree of freedom onlong wavelengths. We then focus on a completely level potential surface, V = V0, extending froma specified exit point φe, where slow roll resumes or inflation ends, to φ → +∞. We compute theprobability distribution in the number of e-folds N required to reach φe which allows for the com-putation of the curvature perturbation. We find that, if the field’s initial velocity is high enough, allpoints eventually exit through φe and a finite curvature perturbation is generated. On the contrary,if the initial velocity is low, some points enter an eternally inflating regime despite the existence ofφe. In that case the probability distribution for N , although normalizable, does not possess finitemoments, leading to a divergent curvature perturbation.

I. INTRODUCTION

The ∆N formalism is a very convenient way to com-pute the curvature perturbation generated during in-flation. Its basic tenet is that quantum fluctuationsstretched to superhorizon scales introduce randomness inthe total number of e-folds at different points in the uni-verse, with this number of e-folds counted from a giveninitial spatially flat time-slice to a given final uniform φtime-slice. This final time-slice is determined by a pre-scribed condition on the scalar field, for example thatslow-roll, and presumably inflation, ends. This differ-ence in the number of e-folds between different spatialpoints directly gives the scalar curvature perturbation.The idea that a time delay encodes the curvature pertur-bation induced by the fluctuations of the inflaton goesback to the early days of inflationary cosmology and wasalready used in some of the pioneering papers on infla-tionary pertubations [1, 2]. The relation of classicalized,super-Hubble modes to a time delay in the dynamics wasclearly explained in [3]. The concept was further for-malized and connected to the conserved gauge invariantcurvature perturbation in [4] and later in [5, 6]. More re-cently, it was re-introduced and elaborated in [7, 8], seealso e.g. [9, 10].

In the usual slow roll scenario, the number of e-foldsbetween the prescribed time slices is dominated by theclassical/deterministic result and random perturbationsare introduced only as a fluctuation of the initial condi-tion in φ, generated when a given mode exits the Hubbleradius. In this regime, the evolution of the probabilitydistribution is dominated by the drift term of the Fokker-Planck equation. However, when the potential is veryflat, as in the case of ultra slow roll [11–15], and the driftterm is small, the slow-roll formula for the curvature per-

turbation cannot be used any more and the e-fold num-ber becomes an essentially stochastic quantity. Whenasking for the time it takes for the field to reach the pre-scribed value φe we are thus facing a first-passage timeproblem in the stochastically evolving system: Given aninitial condition, how many e-folds N are needed for φe

to be reached? The total number of e-folds becomes astochastic quantity described by a probability distribu-tion %(N ), such that %(N )dN is the probability that thefield will reach φe for the first time within the interval[N ,N + dN ) of e-folds [16–21]. As alluded to above, onecan define the first passage with respect to any desirablecondition labelled by φe and defined by a constant fieldhypersurface where φ = φe, such as inflation ending orthe commencement of another distinct phase.

The curvature perturbation generated during a phaseof Ultra Slow Roll (USR) has attracted considerable at-tention recently [22–29] due to the possibility that it leadsto an enhanced curvature perturbation and a related en-hanced primordial black hole production - see [30] for arecent review on cosmological implications of primordialblack holes. In this work we revisit the problem, takingthe scalar sector of gravity fully into account. We placethe computation within the framework of a long wave-length (leading gradient) approximation to the equationsof General Relativity for an inhomogeneous universe: weretain full non-linearities but drop terms that are sec-ond order in spatial gradients and properly take into ac-count the field’s velocity and the corresponding gravita-tional backreaction. Quantum fluctuations are then con-sistently included as a random forcing of the dynamicalequation of the scalar field, a well established approxi-mation for IR quantum fields in inflationary spacetimes[16, 32–37]. We find that imposing the 0i Einstein equa-tion, the GR momentum constraint, leads to the field φ

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being the only dynamical stochastic variable. The field’svelocity is constrained and does not obey an indepen-dent equation involving different stochastic kicks at dif-ferent spatial points, unlike what a naıve “separate uni-verse” argument would imply. This is a non-linear gen-eralization of the linear perturbation theory result fork/aH → 0 [4, 31].

We apply the formalism to a simple problem: the cur-vature fluctuation generated in an extreme version ofUSR where the field is injected at some point φin withvelocity Πin on a totally flat potential V = V0. We findtwo separate regimes, depending on the distance from φin

to the exit point φe and the initial velocity. If this dis-tance is larger than the length of the classical trajectory,corresponding to a low injection velocity Πin, the field ex-periences what we call the stochastic conveyor belt modelfor USR: in some parts of the universe the initial velocityis forgotten and the field explores the infinite semi-lineφ → ∞, never fully reaching φe. The resulting proba-bility distribution for the total number of e-folds N isnormalizable but does not have finite moments, leadingto the infinite inflation observed in [38, 39]. However,if the distance is smaller than the length of the classi-cal trajectory, corresponding to a high injection veloc-ity Πin, graceful exit does occur and inflation eventuallyterminates in all points of the Universe. The curvatureperturbation is then finite and is described by a highlynon-Gaussian probability distribution that we compute.

Obviously, the infinite inflation regime will not bereached in realistic single field models where USR takesplace only on a finite portion of the potential. This paperthen serves an expository function for the developed tech-niques, involving mainly the use of the Hamilton-Jacobiequation for inflationary evolution, the consistent inclu-sion of stochastic fluctuations, and the description of thestochastic conveyor belt mechanism. These techniqueswill be used to analyse more realistic USR potentials ina forthcoming publication [40].

II. LONG WAVELENGTH SCALARPERTURBATIONS

We start by recalling the long wavelength approach ofRef. [4], see also [41], which will take us to the startingpoint of our stochastic analysis. Considering the metricin its ADM parametrization,

g00 = −N2 + γijNiNj , g0i = Ni , gij = γij , (1)

where N and Ni are the lapse function and shift vec-tor respectively, the Einstein equations for gravity plus asingle scalar field φ give the GR energy and momentumconstraints (00 and 0i Einstein equations)

KijKij − 2

3K2 −(3)R+ 16πGε = 0 , (2)

Kji|j −

2

3K|i + 8πGΠφ|i = 0 , (3)

the dynamical equations for the extrinsic curvature ten-sor of the 3-slices Kij = Kij + 1

3Kγij

∂K

∂t−N iK|i = −N |i|i

+N

[3

4KijK

ij +1

2K2 +

1

4(3)R+ 4πGS

], (4)

∂Kik

∂t+N i

|lKlk −N l

|kKil −N lKi

k|l = −N |i|k

+1

3N |l|lδ

ik +N

[KKi

k +(3)Rik − 8πGSik

], (5)

(stemming from the ij Einstein equation) and the equa-tion of motion for the scalar field

1

N

(∂Π

∂t−N iΠ|i

)−KΠ− 1

NN|iφ

|i − φ|iφ|i +dV

dφ= 0 .

(6)In the above, the field momentum Π is defined as

Π =1

N

(∂φ

∂t−N iφ|i

), (7)

the extrinsic curvature 3-tensor is

Kij = − 1

2N

(∂γij∂t−Ni|j −Nj|i

), (8)

and the scalar’s energy density and stress tensor on the3-slices read

ε =1

2

(Π2 + φ|iφ

|i)

+ V (φ) , (9)

and

Sij = φ|iφ|j + γij

(1

2Π2 − 1

2φ|iφ

|i − V (φ)

). (10)

A vertical bar denotes a covariant derivative w.r.t. the3-metric γij which is also used to raise or lower spatialindices.

The approximation we use to study the non-linear longwavelength configurations relevant for inflation is to onlykeep terms containing the leading order in spatial deriva-tives. This is underpinned by the expectation that onscales aL > H−1 the dynamics is dominated by timederivatives such that for any quantity Q the inequality‖e−α∇Q‖ < |∂tQ| will be true, where α denotes thelocal expansion rate (see Eqs. (15) and (17) below), astatement that in inflation is expected to eventually holdfor all scales of interest. Furthermore, to simplify theequations we choose to consider coordinate systems con-structed such that Ni = 0. This gauge choice fixes threegauge degrees of freedom, leaving one gauge function un-fixed; its elimination can be achieved e.g. by furtherchoosing a specific form for the lapse function N . Un-der these assumptions, we get from (5) that the tracelesspart of the extrinsic curvature evolves according to

∂Kik

∂t= NKKi

k . (11)

3

The 3-metric can be further decomposed as

γij = e2αhij (12)

where Dethij = 1 and therefore hij ∂∂thij = 0. We thenhave

K = −3α ≡ −31

N

∂α

∂t, (13)

from where we directly obtain

Kij = Cij(x)e−3α , Tr[Cij(x)] = 0 . (14)

Since during inflation α(t,x) represents the local gener-alization of the number of e-folds, it grows approximatelylinearly in time, and we can take it as a proxy for time ininflation. Therefore, Eq. (14) tells us that the anisotropicexpansion rate Ki

j – which is the non-linear generaliza-tion of the canonical momentum associated with gravita-tional waves – declines extremely rapidly (exponentiallyfast) during inflation. We are thus dynamically led toKi

j = 0 and the most general 3-metric on long wave-lengths can be written as

γij(t,x) = e2α(t,x)hij(x) (15)

with the long wavelength spacetime metric taking theform

ds2 = −N2(t,x)dt2 + e2α(t,x)hij(x)dxidxj . (16)

and the 3-tensor hij is not dynamical in this approxima-tion, at least classically. Furthermore, we restrict thelapse function N(t,x) to vary slowly enough in spacesuch that its spatial gradients can be neglected. Lateron we will consider scalar quantum fluctuations and theaccompanying tensor fluctuations would provide hij witha stochastic source from subhorizon tensor modes enter-ing the long wavelength sector and with an amplitudeset by the uncertainty principle. In this work we focuson the dynamics of the scalar sector of gravity, leavingthat of the stochastic evolution of the tensor sector forfuture study.

Defining the local expansion rate as

H(t,x) ≡ 1

N

∂α

∂t, (17)

we are led to a set of long wavelength equations forthe spatially dependent field φ(t,x) and expansion rateH(t,x) comprising of the two GR constraints: the en-ergy (2) and momentum (3) constraint (from now on weset 8πG = 1 1)

H2 =1

3

(1

2Π2 + V (φ)

)(18)

1 The Newton constant 8πG = 1/M2P can always be recovered in

the equations by noting the canonical dimension of various quan-tities, [H] = 1, [φ] = 1, [8πG] = −2, [Π] = 2, [V (φ)] = 4, [hij ] =0.

and

∂iH = −1

2Π∂iφ , (19)

the evolution of the expansion rate (4),

1

N

∂H

∂t= −1

2Π2 , (20)

as well as the dynamical equations for the scalar field (6–7),

Π =1

N

∂φ

∂t(21)

1

N

∂Π

∂t+ 3HΠ +

dV

dφ= 0 . (22)

Equations (18), (20), (21) and (22) are formally thesame as those of homogeneous cosmology but are validat each spatial point with a priori different values of theinitial conditions for φ and Π. This is what is sometimesreferred to as the “separate universe evolution”. How-ever, not all spatially inhomogeneous initial conditionsare allowed on long wavelengths as they must also sat-isfy the momentum constraint (19). As we will see, thisrestricts the possibility of assigning Π independently ofthe initial value of φ. Indeed, for the constraint (19) tobe respected the local expansion rate H must depend onthe spatial position only through its dependence on thespatially varying φ

H(t,x) = H(φ(t,x), t) (23)

and the field momentum must be given by

Π = −2∂H

∂φ. (24)

Taking the time derivative of (23) and comparing with(20) we immediately find that(

∂H

∂t

)φ

= 0 , (25)

showing that the total spatio-temporal dependence of theexpansion rate is solely determined through its depen-dence on φ(t,x)

H = H(φ(t,x)) , (26)

and the field momentum is therefore given by

Π = −2dH

dφ. (27)

Hence Π is evidently also a function of φ alone, Π =Π(φ(t,x)), with no explicit temporal or spatial depen-dence.

4

When (27) is inserted into the local energy constraint(19) it gives the Hamilton-Jacobi equation for the func-tion H(φ) (

dH

dφ

)2

=3

2H2 − 1

2V (φ) . (28)

Since (28) is a first order differential equation it admitsa family of solutions of the form H(φ, C), with differentsolutions parametrized by an arbitrary constant C. Asolution H = H(φ, C) to (28) along with

dφ

dt= −2N

dH

dφ(29)

provide a complete description and the inhomogeneouslong wavelegth scalar field configuration φ.2 The cor-responding long wavelength metric (16) can then deter-mined via

1

N

∂α

∂t= H(φ) . (30)

It should be pointed out that by taking a φ deriva-tive of (18) one can verify that the long wavelength fieldindeed obeys

1

N

∂

∂t

(1

N

∂φ

∂t

)+ 3H

1

N

∂φ

∂t+dV

dφ= 0 (31)

as expected. Although equations (28) - (30) are identicalto those of homogeneous cosmology, including (31) whichis implied by them, the momentum constraint (19) im-poses that there is no freedom to choose initial values forΠ independently at each spatial point. Let us see why:If (28) is considered without reference to (19), a naıveseparate universe picture would imply that C = C(~x), i.e.for every point ~x on the initial hypersurface there wouldbe a separate integration constant C(~x). This simply en-codes the freedom to choose the initial field momentumat that point independently of φ. However, a spatiallyinhomogeneous C(~x) further leads to

∇H = (∂CH)∇C + (∂φH)∇φ 6= −1

2Π∇φ . (32)

Therefore, as long as ∂CH 6= 0 it follows that ∇C = 0,otherwise the momentum constraint (19) would be vio-lated. This restricts C to be a global constant, meaningthat the momentum Π cannot be chosen arbitrarily ateach point but, instead, all spatial points must be placedalong one and the same integral curve of (28).

2 Equation (28), often referred to as the Hamilton-Jacobi equationin the literature, has been used in [42, 43] to study an alternativeparameterization of possible homogeneous inflationary cosmolo-gies, including USR [12, 44]. We stress that in the present, longwavelength context it describes an inhomogeneous universe aswell, as originally demonstrated in [4].

The restriction ∇C = 0 is a direct consequence ofthe momentum constraint. However, it would not applywhen integral curves of H(φ, C) exist for which ∂CH = 0.In this case setting ∇C 6= 0 would not violate the mo-mentum constraint since a spatially varying value of Cdoes not change the value of H at different spatial points.Such integral curves are attractors of the long wavelengthsystem as one can see by taking a derivative of (28)w.r.t. C which leads to [4, 41]

∂H

∂C∝ a−3 . (33)

This is equivalent to the decay of the second solution to(31) and, as was already pointed out in [9], if this decay-ing mode is neglected the momentum constraint places nofurther restriction on the long wavelength configuration;this is the case for example in Slow Roll inflation. In thecase of motion on a constant potential examined in thiswork, the decaying mode (33) expresses the approach toa static field and de Sitter spacetime - see (38) where theconstant φ0 in that formula plays the role of the generalconstant C discussed here. Including this decaying modeis therefore crucial in studying the slide and slowing downof the field on the constant potential. More generally, theHamilton-Jacobi treatment adopted here allows the ∆Nformalism to be made fully and a priori consistent withthe momentum constraint and include the decaying modewhich is essential in studying Ultra Slow Roll.

We can conclude that on sufficiently large scales, atwhich subleading spatial gradient terms can be neglected,the dynamics of the inhomogeneous configuration can bedescribed solely in terms of φ while the momentum, whenit contributes to the expansion rate, cannot be arbitrarilychosen at different spatial points. The would-be inhomo-geneous degree of freedom is killed by the momentumconstraint (19) on long wavelengths which patches dif-ferent spatial points together. Physically, that meansthat on large scales there is a decaying inhomogeneousmode which is necessarily suppressed by spatial deriva-tives, and hence negligible in the leading gradient expan-sion. We stress that this conclusion goes beyond slowroll and is completely general, relying only on the longwavelength approximation. Note that slow-roll triviallysatisfies the momentum constraint and is an attractor inwhich any dependence of H on C is altogether suppressedexponentially. The absence of a second dynamical inho-mogeneous mode is unique to single scalar inflationarymodels. For a recent study on how the inflaton canonicalmomentum can be excited during inflation through itscoupling to a light spectator scalar field, see Ref. [45].

Before closing this section we note that equations (28),(29) and (30) and the corresponding metric (16) are validfor any choice of time hyper-surfaces (any choice of N)as long as the corresponding spatial coordinate world-lines are constructed orthogonal to the time slices, keep-ing Ni = 0, and if all terms that are second order inspatial gradients are dropped, see [4]. Fer the reader’sconvenience, we recall the demonstration of this fact in

5

Appendix A.

III. HAMILTON-JACOBI SOLUTION IN ULTRASLOW-ROLL

We now apply the above analysis to our extremal USRscenario where the field moves along a very flat and levelpart of the potential where V (φ) ' V0. Equation (28)then reads (

dH

dφ

)2

=3

2H2 − 1

2V0 . (34)

Taking a derivative with respect to φ gives,

dH

dφ

(d2H

dφ2− 3

2H

)= 0 , (35)

and the general solution can be written as

H(φ) = H0 =

√V0

3, or H(φ) = Ae−

√32φ +Be

√32φ

(36)where AB = V0

12 . The constraints on the integration con-stants H0 and AB are obtained by inserting the generalsolutions of (35) into the original Hamilton Jacobi equa-tion (34). It is convenient to redefine A and B as

A =H0

2C =

H0

2e√

32φ0 , B =

H0

2C−1 =

H0

2e−√

32φ0 ,

(37)where C and φ0 are global constants. With these defini-tions in mind, the second solution in (36) becomes,

H(φ) =H0

2

(Ce−√

32φ + C−1e

√32φ)

= H0 cosh

(√3

2(φ− φ0)

), (38)

where we replaced C by φ0 =√

23 ln(C). The correspond-

ing momentum (velocity) is then from (24),

Π = 0 , or Π = −√

6H0 sinh

(√3

2(φ− φ0)

). (39)

The number of e-foldings α, from (17) and using the so-lution for H(φ) (38) becomes

α =

∫HNdt = −1

6ln

[sinh2

(√3

2(φ− φ0)

)]. (40)

The above formulae completely describe the classicalfield evolution along a flat potential V = V0, includingthe gravitational backreaction of a non-zero field veloc-ity. However, they require some clarification with regardsto the exact dynamics they describe. We see that if the

FIG. III.1. The conveyor belt of stochastic Ultra Slow-Roll:Solutions of the HJ equation of motion on a flat potentialV (φ) = V0 are plotted. Inflation ends at φ = 0 in this figure.The blue and red curves are HJ solutions for fields movingto the left (negative initial velocity) while the dashed curveshows a trajectory that started off with positive initial veloc-ity. These evolving trajectories reach the H2

0 = V0/3 surfaceonly asymptotically after an infinite number of e-folds. Alter-natively, the field can remain stationary at some value of φ onone of the H2

0 = V0/3 points - a sample of them is denoted inthe figure by diamonds. Classically, the field does not transi-tion from an evolving state to a stationary state and thereforethe classical phase space is partitioned into these two typesof trajectories. Stochastic fluctuations disintegrate this par-tition: the field can start along one of the HJ trajectories butit is now possible to cross the asymptotic end point due to astochastic jump. This end point acts as a bifurcation pointfor the quantum phase-space, beyond which the field simplydiffuses along the H2 = V0/3 surface. The GR momentumconstraint is still respected since Π = 0 there. The systemthus resembles a jiggly conveyor belt where an initial HJ tra-jectory feeds into a de Sitter stage with successive unimpededdiffusion.

field starts off with a finite velocity, it evolves asymptot-ically towards φ0, which it reaches only after an infiniteamount of e-folds. The precise value of φ0 depends onthe initial velocity imparted on φ as well as its sign: IfΠ(φin) > 0 then φ0 > φ(t) and φ moves asymptotically tothe right towards φ0. If Π(φin) < 0 then φ0 < φ(t) and φmoves asymptotically to the left towards φ0. Of course,if Π(φin) = 0 then the field remains static and these aredegenerate “trajectories” where Π = 0 and H = H0 al-ways. Note that they represent distinct solutions of (28)and the field does not transition from a Π 6= 0 to a Π = 0state during its classical evolution, nor does its velocitychange sign. The constant φ0 is the asymptotic end-pointof each classical trajectory and it parametrizes differentintegral curves of the HJ equation (28); it is identifiedwith the constant C of the general discussion in section2. Note that ∂φ0

H|φ=φ0= 0 and therefore a static field

6

can become inhomogeneous without violating the mo-mentum constraint. Figure 3.1 summarises the differentsolutions. We therefore see that on long wavelengths andfor a flat level potential:

• An evolving (Π 6= 0) long wavelength field configu-ration always tends at asymptotically late times tothe same field value at all spatial points, φ(t,x)→φ0.

• An arbitrary inhomogeneous field configuration isonly allowed for a static field, since ∂φ0

H|φ=φ0= 0

in this case. This is a non-linear version of thegrowing mode of linearized pertubations

As emphasized above, these represent two different solu-tions which classically do not evolve into each other. Inthe following section we discuss how this picture changeswhen quantum effects are included.

IV. STOCHASTIC EVOLUTION: THECONVEYOR BELT OF ULTRA SLOW ROLL

Let us now incorporate a stochastic element in theevolution, modelling as usual quantum fluctuationsstretched to long wavelengths. We do not need to spec-ify the amplitude of the noise terms at this point sowe keep them general for this discussion. Generically,any stochastic ”add-on” to the dynamics, regardless ofthe microscopic origin of the extra noise terms, can bethought of as adding an extra stochastic “kick” to theclassical drift determined by the dynamical equations.Suppose we introduce noise in both the field and its mo-mentum: in a discretized form of the time evolution theirvalues would be updated after a time step ∆t as

∆φ = −2∂H

∂φN∆t+ ξφN∆t ,

〈ξφ(t)ξφ(t′)〉 = Aδ(t− t′) , (41)

∆Π = −3HΠN∆t+ ξΠN∆t ,

〈ξΠ(t)ξΠ(t′)〉 = Bδ(t− t′) . (42)

To be consistent with the constraints, H should be asolution to the Hamilton-Jacobi equation. For the casewe are considering it is given by (38) with either φ > φ0

or φ < φ0 depending on the sign of the velocity - seefigure 3.1.

The energy and momentum constraints are not dy-namical equations and therefore they are not to be ac-companied by some form of an extra stochastic force.As is well known, the constraints are preserved by theclassical dynamical evolution and any consistent stochas-tic extension of the dynamics should also preserve themwhile the stochastic kicks are incorporated - the stochas-tically updated field and momentum should respect themtoo. We can achieve this by elevating the constants ap-pearing in the solution of the HJ equation, whose values

parametrise different possible velocities for fixed field val-ues, to stochastic variables. Let’s now see how this canbe done in our case.

The constant φ0 in (39), characterizing different HJsolutions, can be linked to the velocity since we can write

φ0 = φ−√

2

3arc sinh

(− Π√

2V1/20

), (43)

and the choice of Π for fixed φ is reflected in φ0 which alsodefines the breadth of field values covered by motion onthe flat potential given the initial Π of the field. If manypossible initial conditions for Π are contemplated, thenφ0 defines the different asymptotic resting points corre-sponding to different initial momenta Πin for a given ini-tial value of φ. A stochastic change in Π at fixed φ wouldcorrespond to the field changing the HJ curve along whichit evolves and this can be accommodated by promotingφ0 to a stochastic variable which would change accordingto 3

∆φ0 = ∆φ+∆Π

3H+

1

18H3

∂H

∂φ∆Π2 , (44)

leading to

∆φ0 =NB

18H3

∂H

∂φN∆t+

(1

3HξΠ + ξφ

)N∆t . (45)

At face value this provides a stochastic equation forφ0 which would now take different values at differentspatial points. However, as we stressed above, unless∂H∂φ0

= −∂H∂φ = Π2 = 0, φ0 should only take a global value

if the momentum constraint is to be respected. Thiscannot be accommodated in (45) since, for any choiceof ξφ and ξΠ, φ0 necessarily develops inhomogeneities.Hence, although one could a priory allow for stochasticchanges in the velocity through stochastically jumpingbetween different HJ trajectories on top of the stochasticφ displacement, the momentum constraint prevents thatif Π 6= 0.

We are thus led to conclude that as long as Π 6= 0 thewhole long wavelegth universe can only be located on dif-ferent points of a single HJ trajectory with the followingstochastic equation:

∆φ = −2∂H(φ, φ0)

∂φN∆t+ ξφN∆t ,

〈ξφ(t)ξφ(t′)〉 = Aδ(t− t′) , (46)

with either φ > φ0 or φ < φ0, depending on the fixedsign of the momentum. Once stochastic evolution takesthe field past φ0, memory of the initial velocity is lost

3 Note that we are using Ito’s calculus here [46, 47] and we there-fore keep ∆Π2 terms to follow changes to order ∆t. Other choicesare possible along with corresponding calculi and the results areinvariant since A and B are independent of φ.

7

and it simply diffuses by a free random walk on the flatpotential surface V0 obeying

∆φ = ξφ(t,x)N∆t , 〈ξφ(t)ξφ(t′)〉 = Aδ(t− t′) . (47)

In this regime the field does jump between different HJtrajectories, i.e. different points on the H = H0 sur-face. This is now allowed as these degenerate solutionsare characterized by ∂φ0

H|φ=φ0= Π(φ0) = 0. Hence the

momentum constraint is not violated by the universe oc-cupying different solutions at different spatial points andstochastically jumping between them, becoming a collec-tion of classically static field values that carry no extraenergy.

Note that the quantum fluctuations have a remarkableeffect. The classical phase-space, consisting of the set oftrajectories that solve the HJ equation (28) and whichare shown in figure 3.1, is split into (a) regular (non-degenerate, HJ) trajectories, which are characterized byan initial momentum Πin, the corresponding field value,φin = φ(Πin), and end at φ0 = φ0(Πin) at which Π = 0;(b) degenerate trajectories characterized by Πin = 0 andan arbitrary field value φ0. The quantum phase-spaceis very different however. A typical quantum/stochastictrajectory consists of a classical HJ branch Πin, whichends at φ0 = φ(Πin), supplemented by the set of alldegenerate trajectories, (Π = 0, φ ∈ R). The pointφ0 = φ(Πin) is a bifurcation point, at which the quantumtrajectory splits into two branches: φ > φ0 and φ < φ0,see figure 3.1. This quantum phase space picture resem-bles a conveyor belt for the quantum field which startsat a point on one of the HJ branches, diffuses downwardstowards the bifurcation point at φ0 and then continuesdiffusing along the set of points shown as the horizontalline Π = 0, H = H0 ≡

√V0/3 in figure 3.1.

Despite field fluctuations being generated, the abovepicture does not carry with it a well-defined curvatureperturbation. A corresponding curvature perturbationemerges only when an exit point φe is specified, whereeither inflation ends or another inflationary era followsby exiting the region where V (φ) = V0. If φe lies outsidethe HJ branch, we are faced with the stochastic conveyorbelt and a double first passage-time problem: Firstly totransition from a Π 6= 0 solution onto the H = H0 sur-face (non-stochastic evolution does not allow this) andsecondly to exit the H = H0 region by reaching φe. Ifφe is reached within the HJ branch we have a standardfirst passage-time problem and the conveyor is not op-erational. We analyse these cases in sections V and VIrespectively.

V. THE CASE φe < φ0: USR WITHOUTGRACEFUL EXIT

As we demonstrated above, the gravitationally consis-tent inclusion of velocity to the problem of diffusion ona flat potential leads naturally to a two stage processwhen φe < φ0: 1) All spatial points diffuse along a single

branch of the HJ solution until φ0 is crossed and then2) each point that has crossed φ0 diffuses independentlyalong the level V0 potential. We therefore need to con-struct a first passage time probability distribution for thefirst stage and for that we require the kernel (to whichwe shall also refer to as the propagator) with exit bound-ary conditions at φ0, achieved by setting PHJ(φ0, α) = 0[48]. The probability current at that point then injectsprobability for the second stage of the diffusion - one canthus think of the HJ branch as a “conveyor belt” feedingthe second diffusive process at a single point φ0.

The stochastic equation describing the IR field dynam-ics reads,

dφ = −2∂H

∂φdτ + ξφdτ , (48)

where ξφ is the noise generated by the flow of modes be-tween the UV and IR sectors of the theory. When treatedperturbatively, due to an effectively time-dependent cut-off, the leading order contribution occurs at the treelevel [16] and on super-Hubble scales the noise is, to agood approximation, of Markovian type,

〈ξφ(τ)ξφ(τ ′)〉 = Aδ(τ − τ ′) , (49)

where τ =∫ tN(t, ~x)dt′ denotes a reparametrization in-

variant time. The coupling between the ultraviolet andlong wavelength modes can then be approximated by itstree level expression,

A =(σaH)3

2π2(1− ε)H|φ(τ, k)|2k=σaH

[1 +O(κ2H2)

],

(50)where κ2 = 16πG is the loop counting parameter ofquantum gravity and σ sets the highest (ultraviolet cut-off) energy scale of the long-wavelength theory. For ex-ample, when σ = 1, the highest scale (smallest wave-length) is the Hubble scale, when σ � 1, the highestscale is much smaller than the Hubble scale (or equiva-lently wavelength much longer than H−1). Since therecan be no secular enhancement in the loop correctionsin (50), the loop suppression factor, κ2H2 . 10−12 rep-resents a fair estimate of the accuracy of the stochasticapproximation scheme developed in this work. In orderto estimate the noise amplitude (50), in what follows wework in the approximation ε ≈ 0 and set σ � 1, in whichcase |φ(τ, k)|2 ' H2

0/(2k3) and the noise amplitude (50)

simplifies to,

A ≈ H30

4π2, (51)

which is the approximation we use below. Rigorous proofthat Starobinsky’s stochastic inflation [16] reproduces thecorrect infrared dynamics on de Sitter can be found inRef. [32–37] for interacting scalar field theories and inRef. [50] for quantum scalar electrodynamics. Theseworks demonstrate that stochastic inflation not only re-produces the leading infrared logarithms at each order

8

in perturbation theory, but (when summed up) they alsoreveal what happens at late times in the deep nonper-turbative regime when the large logarithms overwhelmsmall coupling constants, a point first made in [49]. Todirectly compute the curvature perturbation we use thenumber of e-folds α as the time variable.4 This is infact required for consistency with standard cosmologicalpertubation theory, see e.g. [21]. We therefore have thefollowing branches of the evolution:

HJ Branch: The Langevin equation on the HJ branch is

dφ

dα= −2

∂ lnH(φ, φ0)

∂φ+H(φ, φ0)

2πξ(α) (52)

with

〈ξ(α)ξ(α′)〉 = δ (α− α′) (53)

and an absorbing boundary condition at φ = φ0.H(φ, φ0) is the solution to the HJ equation with φ0 deter-mined by the initial velocity of the field. When expressedas a Fokker-Planck equation this implies that the proba-bility density PHJ(φ, α) on the HJ branch obeys,

∂PHJ

∂α= − ∂

∂φ

(−2

∂ lnH

∂φPHJ

)+

1

2

∂2

∂φ2

(H2

4π2PHJ

)(54)

≡ −∂J∂φ

, (55)

to be solved with the boundary condition PHJ(φ0, α) = 0which implies that once a random walker φ among theensemble ventures to φ = φ0, it is removed - see e.g.Ref. [48] for a detailed discussion of this boundary con-dition’s use in exit problems.

H0 (de Sitter) branch: Once the stochastically evolvingfield at a spatial point reaches φ0, it is removed from theHJ branch and is injected into the degenerate V = V0

(de Sitter) branch, where H = H0 =√V0/3. It then

diffuses along the semi-infinite branch φ ∈ [φe,∞) of theflat potential V = V0 according to the Langevin equation,

dφ

dα=H0

2πξ(α) (56)

again with

〈ξ(α)ξ(α′)〉 = δ (α− α′) , (57)

where now the influx from the HJ branch must also beaccounted for. When the exit point φe of the V = V0

branch is reached inflation may end, for example by en-tering a non-slow roll region or by instant reheating, or

4 We are therefore using a uniform expansion gauge in perturba-tion theory terminology. This is not fully equivalent to choosingspatially flat time slices but, as pointed out in [4], volume pre-serving shape deformations are not interesting dynamically inour approximations. See also appendix A on this point.

the field may enter a subsequent slow roll phase. In ei-ther case, the quantity of interest is the number of e-foldsuntil φe is reached which is a stochastic quantity.

We can write the probability distribution for φ on theH0 branch as

PV0(φ) = PD(φ) + P? δ (φ− φ0) (58)

where PD is that part which has diffused along the V =V0 surface while P? denotes the probability at φ0 leakingin from the HJ branch. Its contribution to the Fokker-Planck equation on the V = V0 branch can be computedas follows: in a time interval between α and α + dα theamount of random walkers flowing in from the HJ branchis

dP? =

∫ ∞φ0

[PHJ (φ, α+ dα)−PHJ (φ, α)

]dφ (59)

Therefore,

∂P?∂α

=

∫ ∞φ0

∂PHJ

∂αdφ = J(φ0, α) . (60)

and the Fokker-Planck equation for PV0can then be writ-

ten as

∂PV0

∂α=

1

2

∂2

∂φ2

(V0

12π2PV0

)+ J(φ0, α)δ (φ− φ0) , (61)

where, recalling that PHJ(φ0) = 0 and ∂φH(φ0) = 0,

J(φ0, α) =V0

24π2

∂PHJ

∂φ

∣∣∣φ0

. (62)

The probability distribution for the number of e-foldsit takes for the field to reach φe can be obtained fromknowledge of PD by noting that once the random walkerhas been injected into the V0 branch and has starteddiffusing, the probability it hasn’t yet crossed φe by thetime of N e-folds is the same as that of inflation lastinglonger than N e-folds:

Prob(Inflationary duration > N ) =

∞∫N

%(α)dα

=

∞∫φe

PD(φ,N ) dφ (63)

where we denoted the probability that inflation lasts(more precisely φe is reached) between α and α + dαe-folds by %(α). Therefore

%(N ) = − ∂

∂N

∞∫φe

PD(φ,N ) dφ . (64)

Using (58), (60) and (61) we obtain simply

%(N ) =V0

24π2

∂PV0(φ,N )

∂φ

∣∣∣φe

. (65)

9

A. Computing PHJ

In order to obtain the current flowing into the V0

branch from (62) we first need to compute PHJ, the prob-ability distribution on the HJ branch. To obtain sim-ple analytic expressions, we will make the approximationthat φ is close to φ0 on the HJ branch, corresponding toa small initial velocity. This is justified since

φin−φ0 = arc sinh

(− Πin√

2V1/20

)' − Πin√

2V1/20

� 1 (66)

assuming that the field enters the USR regime from aprevious slow roll phase. We will tackle the more generalproblem in an upcoming publication [? ]. We thereforetake the HJ branch stochastic dynamics to be (see 38)

dφ

dα' −3 (φ−φ0) +

H0

2πξ(α) (67)

with an exit boundary conditions at φ0. Setting χ =√12πH0

(φ− φ0) the corresponding Fokker-Planck equa-

tion (54) for the probability density PHJ(χ, α) reads,

∂PHJ

∂α' 3

∂

∂χ(χPHJ) +

3

2

∂2PHJ

∂χ2. (68)

Writing

PHJ(χ, α) = Ce32α−

12χ

2

Ψ(χ, α) , (69)

where C is a constant independent of α and χ but de-pendent on the choice of the initial state, Ψ(χ, α) obeys

− 1

3

∂Ψ

∂α=

1

2

(− ∂2

∂χ2+ χ2

)Ψ , (70)

and the problem reduces to the quantum mechanical ker-nel for the simple harmonic oscillator (SHO), with amass m and frequency ω given by mω → ~H2

0/(12π2))in imaginary time α = iωt/(3~) (or t = −3i~α/ω), seee.g. [51]. The free propagator, also known in the lit-erature on stochastic processes as the Mehler heat ker-nel [52], is given by

KM (χ, α;χin, αin) =1√

2π sinh[3(α−αin)]exp

(−coth[3(α−αin)](χ2+χin

2)

2+

χχin

sinh[3(α−αin)]

)(71)

which for small time intervals tends to

limα→αin

KM (χ, α;χin, αin) = δ(χ−χin) . (72)

Since a random walker is ‘removed’ upon reaching χ =0 (φ = φ0), for the problem at hand we do not requirethe free, but rather the absorbtive kernel. Due to thesymmetry of the effective potential in which the dynamicstakes place, it can be obtained from the full kernel (71)by adding to it a free mirror kernel at −χ, giving

Ψ(χ, α) = KM (χ, α;χin, αin)−KM (−χ, α;χin, αin)

=

√2

π

exp(− 1

2 coth[3(α−αin)](χ2+χin2))√

sinh[3(α−αin)]

× sinh

(χχin

sinh[3(α−αin)]

)(73)

which ensures the correct boundary condition is satisfied.The properly normalized PHJ is then obtained from (69),

PHJ(ϕ, α) =6π√V0

exp

(3

2(α−αin)− 1

2(χ2−χ2

in)

)Ψ(χ, α) ,

(74)where C in Eq. (69) is chosen such that in the limit α→

αin reduces to 5

PHJ(ϕ, α→ αin) =6π√V0

δ(χ−χin) = δ(φ−φin) , (75)

for (χ ≥ 0, χin > 0). A general probability distribu-tion on the HJ branch can be obtained by convolving theabsorbtive kernel (73) with the initial probability distri-bution.

The injected current (62), J = [√V0/(4π)]∂χPHJ|χ→0,

into the flat V0 branch is obtained by taking a derivativeof (74),

J(α) =6π[

2π sinh[3∆α]]3/2

× exp

[3

2∆α− 1

2

(coth[3∆α]−1

)χ2

in

]χin , (76)

which rises at early times ∆α = α−αin � 1 as,

J(α)|∆α�1 =χin√

6π(∆α)32

e−χ2

in6∆α , (77)

5 Strictly speaking, in the limit α → αin the probability den-sity (74) reduces to δ(χ−χin) + δ(χ+χin). Since the domainof validity of (73) is the HJ branch on which χ ≥ 0, the seconddelta function is discarded.

10

a

J(a)

FIG. V.1. The current J(α) injected from the HJ branch intothe flat branch at φ = φ0 for χin = 0.5, 1, 1.5 (from leftmostto rightmost curve).

whereas at late times, when ∆α = α−αin � 1, it decaysexponentially,

J(α)|∆α�1 =6√πχine

−3∆α +O(e−9∆α

). (78)

The current J(α) for three different values of χin can beseen in figure V A. The current increases from zero att = 0 (α = αin), peaks and then decays exponentially as∝ e−3(α−αin), see (77) and (78).

B. Computing PV0

Assuming that the initial field distribution lies entirelyat the HJ branch, equation (61) must be supplemented

by the initial condition P (χ, 0) = 0 and the solution cantherefore be written as

PV0(φ, α) =

∫ α

αin

duGe(φ− φ0, α− u)J(u) , (79)

where Ge (φ− φ′, α− α′) is the diffusive Green functionwith exit boundary condition at φe, also known as theabsorptive kernel. As above, it is straightforwardly con-structed from the well known unrestricted diffusive kernelalong an infinite interval

G (φ, φ′,∆α) =

√2π

H20 ∆α

exp

(− 2π2

H20 ∆α

(φ−φ′)2

)(80)

where ∆α = α−α′, by subtracting the same kernel butwith φ′ reflected on φe: φ→ 2φe − φ, giving

Ge (φ, φ′,∆α) =

√2π

H20 ∆α

[exp

(− 2π2

H20 ∆α

(φ−φ′)2

)

− exp

(− 2π2

H20 ∆α

(2φe− φ−φ′)2

)]. (81)

This imposes the correct boundary conditions,Ge (φe, φ

′,∆α) = 0 and Ge (φ, φ′,∆α→ 0) = δ(φ−φ′)for φ, φ′ ≥ φe. The limits of integration in (79) aredetermined by imposing that no current can be sourcedbefore the beginning of inflation at αin (lower limit) andthat no current can be sourced in the future of α (upperlimit).

To compute the probability distribution for the field onthe flat branch we use the convolution integral (79) withthe absorbtive kernel is (81) and the injected current (76)to obtain,

PV0(φ, α) =

3χin

H0

∫ α

0

du√α−u

[exp

(− χ2

6(α−u)

)−exp

(− (χ−2χe)

2

6(α−u)

)]exp

[32u−

12

(coth(3u)−1

)χ2

in

][sinh(3u)

]3/2 , (82)

where χ =√

12π(φ − φ0)/H0, χe =√

12π(φe−φ0)/H0

and we set, for simplicity, αin = 0 6.

6 One can always recover the dependence on αin by noting thatthe integral (82) is a function of α−αin

C. Probability density for the e-fold number

From PV0we can directly compute the e-fold proba-

bility density using (65)

%(N ) =

√3

2π(−χeχin)

×∫ N

0

duexp

[− χ2

e

6(N−u) + 32u−

12

(coth(3u)−1

)χ2

in

][

(N − u) sinh(3u)]3/2 (83)

where we note that χe < 0 by definition. Although theabove integrals cannot be evaluated analytically, an ap-proximate evaluation of (82) can be performed by noting

11

Ν

Ν

FIG. V.2. The probability distribution %(N ) defined by (83)for χe = −1 and χin = 0.5 (leftmost, blue curve) or χin = 1.5(rightmost, red curve). The dashed-dotted line indicates the

asymptotic % ∝ N−3/2 behavior for large N , see (85). Thisdeep non-Gaussian tail is responsible for eternal inflation.

that the dominant dependence on u sits in the exponentand the integral can be well approximated by a steepestdescent method presented in Appendix B. There is a verysimple case, namely if the integral is dominated by u� 1and if α� 1, then it evaluates to,

PV0(φ , α) ≈√

12π

H0

e−χ2in/2

√6π∆α

×[exp

(− χ2

6∆α

)− exp

(− (χ−2χe)

2

6∆α

)], (84)

which is, up to the factor e−χ2in/2, equal to the absorb-

tive kernel Ge(φ−φ0; ∆α) (∆α = α−αin) in Eq. (81).Therefore

%(N ) ≈√

6

πe−

χ2in2e−

χ2e

6N

N 3/2(85)

We see that although %(N ) is normalisable, the proba-bility distribution does not decay fast enough as φ→∞and therefore all moments are infinite: 〈Nn〉 = ∞ forn ≥ 1. A numerical evaluation of the probability density%(N ) is plotted in figure 5.2. For large N the distribu-tion tends to an ∝ N−3/2 decay, which is in agreementwith our analytic estimate. This reflects the fact that ifthe precipice signified by φe is beyond φ0, the field settlesinto free diffusion along the half-line towards φ → ∞, asituation termed infinite inflation in [38, 39].

The endless diffusion towards φ→∞ would of coursenot occur if the field was injected into the de Sitter branchfrom a prior slow-roll regime. We will deal with this inmore detail in [? ] where more complete models arestudied. A simple way to regulate this infinite diffusionwould be to erect a reflecting wall at, or close to φin. In[23] this is shown to indeed lead to a distribution withfinite moments and hence a finite curvature perturbation.VI. THE CASE φe > φ0: USR WITH GRACEFUL

EXIT

We saw in the previous section that if the initial ve-locity of the field does not suffice to carry it beyondφe (φe < φ0), eternal inflation sets in on the semi-line[φe,∞) and the curvature perturbation is infinite, as sig-nified by the divergence of all moments of N . We showin this section that this is not true when the exit pointφe occurs on the HJ branch, i.e. before the asymptoticpoint φ = φ0 at which the classical trajectory of φ wouldterminate. In other words we now assume that,

φe ≥ φ0 , (86)

and show that this model of inflation exhibits a grace-ful exit. The probability density PHJ(φ, α) is then ofthe form (74), but with Ψ(χ, α) given by the absorbtivekernel mirrored at χe, see Eq. (73),

Ψ(χ, α) = KM (χ, α;χin, αin)−KM (2χe − χ, α;χin, αin)

=1√

2π sinh[3(α−αin)]

{exp

(−1

2coth[3(α−αin)](χ2+χin

2)+χχin

sinh[3(α−αin)]

)

− exp

(−1

2coth[3(α−αin)]

[(2χe−χ)2+χin

2]+

(2χe−χ)χin

sinh[3(α−αin)]

)}, (87)

The probability %(N ) in (64) that inflation ends in the interval [N ,N + dN ) of e-folds is then,

%(N ) = − ∂

∂N

∫ ∞φe

dφPHJ(N , φ) (88)

= − ∂

∂N

{1

2erfc

[√1+n

(χe−e−3∆Nχin

)]− exp

[χ2

in−χ2e−(χin−e−3∆Nχe)2

]1

2erfc

[√n(χin−e−3∆Nχe

) ]}(89)

=3√n√π

[2(1+n)χin−(3+2n)e−3∆Nχe

]exp[−(1+n)

(χe−e−3∆Nχin

)2]−3e−3∆Nχe

[χin−e−3∆Nχe

]e(χ2

in−χ2e)−(χin−e−3∆Nχe)

2

erfc[√

n(χin−e−3∆Nχe

) ], (90)

12

χe=0, χin=1

χe=0, χin=1.5

χe=0.3, χin=1

χe=0.3, χin=2

0.2 0.4 0.6 0.8 1.0 1.2 1.4N

1

2

3

4

ρ(N)

FIG. VI.1. The probability distribution (88–90) in the USRmodel with graceful exit as a function of the number of e-folds ∆N = N −Nin with Nin = 0 and given in Eq. (90) forfour choices of (χe, χin): (0, 1) (solid black); (0, 1.5) (solidorange); (0.3, 1) (long dashes) and (0.3, 2) (short dashes).We see that for larger χe (χin) inflation gets shorter (longer),which is what as one would expect.

where n = 1/(e6∆N −1), ∆N = N− Nin and erfc(z) =

1− erf(z) = 2√π

∫∞ze−t

2

dt is the complementary error

function. The distribution %(N ) in (88–90) is plotted infigure VI.1 for a few selected values of χe and χin. Thedistribution is again strongly non-Gaussian, however atlarge N it falls-off exponentially as ∝ e−3N , such thatthe moments of the curvature perturbation are all finite,implying that inflation terminates. The first term afterthe curly bracket in (89) is the standard result for theprobability that the particle is located anywhere at χ >χe, and it approaches one when χe → −∞, as it should,while the second term reduces the probability due to theabsorbtive boundary condition at φ = φe, where inflationends.

Eqs. (88–90) contain a complete information for theprobability distribution of the number of e-folds in thissimple model (where we assumed a small initial momen-tum, which allowed us to linearize in φ− φ0 in (67)). Toget a better understanding of ρ(N ) in (88), we shall nowcalculate the first few moments of the number of e-folds,

〈N k〉 =

∫ ∞αin

N k%(N ) dN , (k = 0, 1, 2, · · · ) . (91)

Let us first look at the zeroth moment,

〈1〉 =1

2

[1 + erf[χe] + e−χ

2e

](χe ≥ 0) . (92)

When 0 ≤ χe . 1 this is, as one would expect, of theorder one. One can account for the fact that (92) is notexactly equal to one by dividing 〈Nn〉 by 〈1〉. The mo-ments of N are considerably more difficult to calculate,and therefore in what follows for simplicity we considerthe case, χe = 0 (φe = φ0). Then the probability distri-bution (88–90) reduces to,

%(N )dN =6√n√π

(1+n)χine−nχ2in dN . (93)

It pays off to convert this into the probability per unitdn = −6n(1 + n)dN ,

G(n)dn =χin√πn

e−nχ2indn , (94)

such that the k-th moment in (91) gives,

〈N k〉 =2√π

∫ ∞0

dy e−y2

[1

6ln

(1 +

χ2in

y2

)]k, (95)

where (k = 0, 1, 2, · · · ) and we used y =√nχin, assuming

that αin = 0. Furthermore, it is useful to calculate howthe number of e-folds fluctuates around its mean value,〈N〉,

〈(∆N )n〉 =

n∑k=0

(−1)k(n

k

)〈N k〉〈N〉n−k . (96)

where(nk

)= n!/[k!(n− k)!] is the binomial coefficient.

The first moment in (96) can be expressed in terms ofa generalized hypergeometric function,

〈N〉 =π

6erf(iχin)−χ

2in

3×2F2

({1, 1},{3

2, 2}, χ2

in

).(97)

The higher moments are harder to evaluate analytically.Nevertheless, one can show that the following confluenthypergeometric function generates all the moments,

G(α, χin) =2√π

∫ ∞0

dy e−y2

(1 +

χ2in

y2

)α6

=1√π

Γ

(1

2− α

6

)× U

(−α

6,

1

2, χ2

in

)= 1F1

(−α

6;

1

2;χ2

in

)− 2

Γ(

12 −

α6

)Γ(−α6) (χ2

in)1/2

× 1F1

(1

2− α

6;

3

2;χ2

in

)(98)

in the sense that

〈N k〉 =

(∂k

∂αkG(α, χin)

)α=0

(k = 1, 2, 3, · · · ) ,

(99)where U denotes the confluent hypergeometric function.In figure VI.2 we show the first few moments in Eq. (95)and their fluctuations around the mean 〈N〉 defined inEq. (96). For simplicity we choose χe = 0 and plot ourresults as a function of χin. We see that the distributionis highly non-Gaussian, which is one one of the main re-sults of this work. Because we have calculated G(α) withthe assumption of small φ−φ0, in figure VI.2 we plot theresults only for χin < 1. The principal conclusion is thatthe non-Gaussianities produced in USR are quite largeand grow with χin, or the length of the USR supportingpotential segment. On the other hand, from figure VI.1we see that a larger χin implies a larger average num-ber of e-folds of USR 〈N〉, from which we conclude that

13

0.2 0.4 0.6 0.8 1.0χin

0.1

0.2

0.3

0.4

0.5<Nn>

<N>

<N2>

<N3>

<N4>

<N5>

0.2 0.4 0.6 0.8 1.0χin

0.1

0.2

0.3

0.4

<(ΔN)n>

<(ΔN)2>

<(ΔN)3>

<(ΔN)4>

<(ΔN)5>

FIG. VI.2. The first few moments of the number of e-foldsfor χe = 0 as a function of χin in our simple USR model. Weshow both the moments of N defined in (91) and (95) (upperpanel), as well as their fluctuation from the mean, 〈(∆N )n〉,defined in (96) (lower panel).

a longer USR phase generates larger non-Gaussianities.This observation can be of crucial importance for thegeneration of primordial black holes. These results are inbroad agreement with the findings of [23] and it wouldbe interesting to make a more quantitative comparison,recalling that we have fully and consistently included thegravitational effects of the field’s velocity.

VII. CONCLUSION AND DISCUSSION

In this paper we established a consistent formalism fordescribing the quantum evolution of the large scale cur-vature perturbation, generated during inflation on veryflat portions of the potential V (φ), fully taking into ac-count the scalar gravitational back-reaction and the fi-nite classical velocity for the field. This was achieved bycombining a long wavelength approximation to the Ein-stein equations with the stochastic picture of inflation-ary quantum fluctuations. We found that the 0i Einsteinequation, usually neglected in the widely used “separateuniverse” approach, leads to a single stochastic equationfor the scalar field but not its velocity, the latter be-ing fully determined by the former even beyond slow rollthrough a unique solution to the Hamilton-Jacobi equa-tion (28).

We then focused on a completely level potential V = V0

where inflation occurs in an ultra slow roll (USR) regime.We assumed that φ ∈ [φe,+∞) and that the field is in-jected with some finite velocity Πin at φin. We showedthat on large (super-Hubble) scales USR is a phase spaceattractor, in the sense that gravitational constraints fully

fix the field velocity in terms of the field φ, up to a globalconstant φ0 determined by the initial velocity and mark-ing the end point towards which the classical field evo-lution asymptotes. This is accurate up to small, expo-nentially decaying gradient corrections, which are highlysuppressed, and thus completely irrelevant on very largescales. The value φe demarcated an exit point whereinflation either ends or the field enters into another re-gion of the potential, presumably one supporting slowroll. The stochastic number of e-folds required to reachφe directly gives the curvature perturbation.

The inflaton dynamics depends crucially on the dis-tance between the entry and exit points |φin − φe| andon the initial field velocity Πin. As we argue in sec-tion VI, if |φin − φ0| > |φin − φe| the field performs agraceful exit, with φe being eventually reached at all spa-tial points. If, on the other hand, |φin − φ0| < |φin − φe|,the quantum phase space becomes larger than the clas-sical one such that USR proceeds in two distinct phases,discussed in detail in section V. When the quantum par-ticle reaches the end point φ0 of the classical trajectory,it will start diffusing along the set of classical trajecto-ries marked by 〈Π〉 = 0 and arbitrary φ, implying thatthe point φ0 acts as a bifurcation point of the quantumphase space, at which the quantum trajectory splits intotwo branches, see figure 3.1. Consequently, a conveyorbelt picture of the quantum particle phase space emergesand leads to a phase where some random walkers exitbut most are trapped in an eternal de Sitter epoch asthe field freely diffuses towards φ → +∞. While in theformer case a well defined probability distribution for thecurvature perturbation emerges, in the latter, althoughnormalizable, the distribution has no finite moments indi-cating an infinite curvature perturbation. This behaviorof course depends on there not being a barrier in reach-ing φ → +∞, a situation not valid in more completeinflationary models.

This is a preliminary study in many respects. In a real-istic inflationary model, the flat potential portion will befinite and even when |φin − φe| is large and the conveyorbelt is operational, the field’s diffusion towards large val-ues will be halted, although it may still lead to a greatlyenhanced curvature perturbation. Furthermore, in pass-ing to equation (67) we linearized in the field pertur-bation φ − φ0, which is equivalent to assuming a smallinitial field velocity Πin. This was done for simplicity andto obtain the semi-analytic results presented here but isnot necessary. This paper was largely expository of themethods developed and we will return with a more gen-eral treatment and more realistic USR models in a forth-coming publication [? ]. Finally, we dropped the tensormodes which are non-dynamical classically. This state-ment will no loger hold when their quantum fluctuationsare taken into account. We reserve a more sophisticatednon-liner treatment of the IR stochastic tensors for thefuture.

Acknowledgements:

14

GR acknowledges partial supported by the STFC grantST/P000371/1 – Particles, Fields and Spacetime. TPacknowledges the D-ITP consortium, a program of theNWO that is funded by the Dutch Ministry of Education,Culture and Science (OCW). We would like ot thank V.Vennin for very useful discussions and the anonymousreferee for insightful comments who allowed us to clarifyapproximations behind our computations.

APPENDIX A: CHANGING THE TIME-SLICINGON LONG WAVELENGTHS

In this appendix, following [4], we recall that underchanges of the time hypersurfaces t → T (t,x) the longwavelength equations (17), (20), (21) and (22) remaininvariant and the long wavelength spatial metric (15) re-tains its form. These statements are valid up to termswhich are second order in spatial gradients and are there-fore dropped within the long wavelength approximation.

Starting with coordinates (t, xi), consider a change inthe choice of constant time hyper-surfaces (the spacetimetime-slicing) defined by a new time coordinate T (t, xi).To keep Ni = 0 in the new coordinate system, new spatialcoordinates Xi must also be chosen which are orthogonalto the T = const surfaces. We now examine how suchthe transformation between the old and new coordinatescan be obtained.

Given the new time surface T (t, xi), a set of spatialcoordinates Xi is chosen on a T = T0 hypersurface andthen orthogonally projected to thread all other T = consthypersurfaces and labelling spatial coordinates in themtoo. Along constant Xi curves the old coordinates xµ willchange as dxµ = T ,µds where s is an arbitrary parameter.Along such lines, T will change as

dT = T,µdxµ = T,µT

,µds (100)

which implies (∂xµ

∂T

)Xj

=T ,µ

T,αT ,α(101)

which defines 4 of the 16 components of the transforma-tion matrix between the old and new coordinates. To de-termine the 12 remaining components consider the trans-formation matrix

Bµk =

(∂xµ

∂Xk

)T

(102)

which should be chosen such that

T,µBµk = 0 (103)

in order to keep gTXi = 0. If condition (103) is satisfiedon the T = T0 hypersurface it will always be satisfied.

This can be seen by taking the T derivative of Bµk to find(∂Bµk∂T

)Xj

=

(∂

∂Xk

(∂xµ

∂T

)Xj

)T

=

(∂

∂Xk

(T ,µ

T,αT ,α

))T

=Bνk

(T ,µ

T,αT ,α

),ν

(104)

In turn, this relation can be used to show that

∂T (T,µBµk ) = 0 (105)

and hence that Ni is kept zero on all T time-slices in the(T,Xj

)coordinates.

From (103) we have

B0k = −B

ikT,iT,0

(106)

which, when substituted in (104) gives(∂Blk∂T

)Xj

=

[(T ,l

T ,aT,a

),m

− T,mT,0

(T ,l

T ,aT,a

),0

]Bmk

(107)The r.h.s. is second order in spatial gradients and isdropped within our approximation scheme, implying thatto this order in the gradient expansion Blk is independentof T

Blk ≡ Blk(X). (108)

On the other hand, by integrating xj along a line of con-stant Xj using (101), we obtain

xj = f j(X) +

∫T ,j

T,0T ,0dT (109)

This is consistent with (108); a derivative of the secondterm w.r.t. Xj involves two spatial gradients ∂/∂xi ascan be seen by using (106). Furthermore, any functionevaluated at xi will read

g(xi) = g

(f i(X) +

∫T ,j

T,0T ,0dT

)' g(f i(X)) + g,i

∫T ,j

T,0T ,0dT . (110)

Hence, within our approximations

g(xi) = g(f i(X)) = g(Xi) (111)

and the 3-metric (15) in the (T,x) coordinates reads

γl′k′ = e2α(t,x)Bll′(X)Bkk′(X)hlk(x)

= e2α(t(T ),X)Bll′(X)Bkk′(X)hlk(X) (112)

i.e it again has the form of a locally defined conformalfactor times a time independent 3-metric which is only afunction of the 3 new spatial coordinates Xi. The sim-plest choice for the new spatial coordinates is of coursef j(X) = Xj .

15

Regarding the dynamical equations in the new time T ,we note that for any time dependent quantity Q(

∂Q

∂T

)Xj

=1

T,0

(∂Q

∂t

)xj

+T ,k

T,0T ,0

(∂Q

∂xk

)t

. (113)

Dropping the second term on the r.h.s. as second order inspatial gradients and noting that the two lapse functions,Nt and NT , associated with the time coordinates t andT respectively are related by Nt = NT ∂T/∂t, we have

1

NT

(∂Q

∂T

)Xj

=1

Nt

(∂Q

∂t

)xj. (114)

up to second order in spatial gradients. This can be usedto show the invariance of the long wavelength dynamicalequations under changes of the time slicing.

APPENDIX B: STEEPEST DESCENT FOR PV0

The integral in (82) is dominated by the dependenceon u in the exponent, which diverges in both limits of in-tegration, and hence is dominated by some intermediateu, at which the function in the exponent minimizes. Tostudy the integral in more detail, we write it in the form,

PV0 =3χin

H0[I(χ, α)−I(χ−2χe, α)] , (115)

I(χ, α) =

∫ α

0

e−S(χ,u)du (116)

where

S(χ, u) =χ2

6(α−u)+

1

2ln (α−u)− 6u

−3

2ln(2) + n(6u)χ2

in −3

2ln[n(6u)] , (117)

and where n(x) = 1/(ex−1) is the Bose-Einstein functionof its argument. The integral (116) is then performed byexpanding S(χ, u) in (117) around the local minimum u0

(at which S′0 ≡ [∂uS(χ, u)]u=u0= 0) as,

S(u) ≈ S(u0) +1

2S′′0 (u−u0)2 +O

((u−u0)3

), (118)

where S′′0 = [∂2uS(χ, u)]u=u0

. Upon dropping the higherorders O

((u−u0)3

), the integral (116) becomes simple to

evaluate,

I(χ, α) =

√π

2S′′0

[Erf

(√S′′02

(α−u0)

)+ Erf

(√S′′02u0

)](119)

where the result is meaningful if S′′0 > 0 . To completethe evaluation, we need u0 and S′′0 , and hence we needthe first and second derivative of (117),

S′0 =χ2

6(α−u)2− 1

2(α−u)− 6n(n+1)χ2

in + 3+9n

= 0 (120)

S′′0 =χ2

3(α−u)3− 1

2(α−u)2

+36n(n+1)(n+2)χ2in−54n(n+1) , (121)

where we made use of ∂un(6u) = −6n(n+1), ∂2un(6u) =

36n(n+1)(n+2). u0 is found by setting S′0 = 0 in (120).If α� u, the problem of solving (120) reduces to findingthe positive root of a quadratic equation in n = n(6u),which is easily solved for n0 ≡ n(6u0), and hence also foru0.

[1] S. W. Hawking, “The Development of Irregularities in aSingle Bubble Inflationary Universe,” Phys. Lett. 115B(1982) 295.

[2] A. A. Starobinsky, “Dynamics of Phase Transition in theNew Inflationary Universe Scenario and Generation ofPerturbations,” Phys. Lett. 117B (1982) 175.

[3] A. H. Guth and S. Y. Pi, “The Quantum Mechanics ofthe Scalar Field in the New Inflationary Universe,” Phys.Rev. D 32 (1985) 1899.

[4] D. S. Salopek and J. R. Bond, “Nonlinear evolution oflong wavelength metric fluctuations in inflationary mod-els,” Phys. Rev. D 42 (1990) 3936.

[5] M. Sasaki and E. D. Stewart, “A General analytic for-mula for the spectral index of the density perturbationsproduced during inflation,” Prog. Theor. Phys. 95 (1996)71 [astro-ph/9507001].

[6] M. Sasaki and T. Tanaka, “Superhorizon scale dynamicsof multiscalar inflation,” Prog. Theor. Phys. 99 (1998),763-782 [arXiv:gr-qc/9801017 [gr-qc]].

[7] D. H. Lyth, K. A. Malik and M. Sasaki, “A General proofof the conservation of the curvature perturbation,” JCAP0505 (2005) 004 [astro-ph/0411220].

[8] D. H. Lyth and Y. Rodriguez, “The Inflationary predic-tion for primordial non-Gaussianity,” Phys. Rev. Lett.95 (2005) 121302 [astro-ph/0504045].

[9] N. S. Sugiyama, E. Komatsu and T. Futamase, “δNformalism,” Phys. Rev. D 87 (2013) no.2, 023530[arXiv:1208.1073 [gr-qc]].

[10] J. Garriga, Y. Urakawa and F. Vernizzi, “δN formalismfrom superpotential and holography,” JCAP 02 (2016),036 [arXiv:1509.07339 [hep-th]].

[11] N. C. Tsamis and R. P. Woodard, Phys. Rev. D 69 (2004)084005

[12] W. H. Kinney, “Horizon crossing and inflation with largeeta,” Phys. Rev. D 72 (2005) 023515

[13] M. H. Namjoo, H. Firouzjahi and M. Sasaki, “Violationof non-Gaussianity consistency relation in a single fieldinflationary model,” EPL 101 (2013) no.3, 39001

16

[14] J. Martin, H. Motohashi and T. Suyama, “Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Re-lation,” Phys. Rev. D 87 (2013) no.2, 023514

[15] K. Dimopoulos, “Ultra slow-roll inflation demystified,”Phys. Lett. B 775 (2017) 262

[16] A. A. Starobinsky, “Stochastic De Sitter (inflationary)Stage In The Early Universe,” Lect. Notes Phys. 246(1986) 107.

[17] D. S. Salopek and J. R. Bond, “Stochastic inflation andnonlinear gravity,” Phys. Rev. D 43 (1991) 1005.

[18] K. Enqvist, S. Nurmi, D. Podolsky and G. I. Rigopou-los, “On the divergences of inflationary superhorizon per-turbations,” JCAP 0804 (2008) 025 [arXiv:0802.0395[astro-ph]].

[19] T. Fujita, M. Kawasaki, Y. Tada and T. Takesako, “Anew algorithm for calculating the curvature perturba-tions in stochastic inflation,” JCAP 1312 (2013) 036[arXiv:1308.4754 [astro-ph.CO]].

[20] T. Fujita, M. Kawasaki and Y. Tada, “Non-perturbativeapproach for curvature perturbations in stochastic δNformalism,” JCAP 1410 (2014) 030 [arXiv:1405.2187[astro-ph.CO]].

[21] V. Vennin and A. A. Starobinsky, “Correlation Functionsin Stochastic Inflation,” Eur. Phys. J. C 75 (2015) 413[arXiv:1506.04732 [hep-th]].

[22] C. Germani and T. Prokopec, “On primordial black holesfrom an inflection point,” Phys. Dark Univ. 18 (2017) 6doi:10.1016/j.dark.2017.09.001 [arXiv:1706.04226 [astro-ph.CO]].

[23] C. Pattison, V. Vennin, H. Assadullahi and D. Wands,“Quantum diffusion during inflation and primordial blackholes,” JCAP 1710 (2017) 046 [arXiv:1707.00537 [hep-th]].

[24] C. Pattison, V. Vennin, H. Assadullahi and D. Wands,“Stochastic inflation beyond slow roll,” JCAP 1907(2019) 031 [arXiv:1905.06300 [astro-ph.CO]].

[25] M. Biagetti, G. Franciolini, A. Kehagias and A. Riotto,“Primordial Black Holes from Inflation and QuantumDiffusion,” JCAP 1807 (2018) 032 [arXiv:1804.07124[astro-ph.CO]].

[26] J. M. Ezquiaga and J. Garcıa-Bellido, “Quantumdiffusion beyond slow-roll: implications for primor-dial black-hole production,” JCAP 1808 (2018) 018[arXiv:1805.06731 [astro-ph.CO]].

[27] D. Cruces, C. Germani and T. Prokopec, “Failure of thestochastic approach to inflation beyond slow-roll,” JCAP1903 (2019) 048 [arXiv:1807.09057 [gr-qc]].

[28] H. Firouzjahi, A. Nassiri-Rad and M. Noorbala,“Stochastic Ultra Slow Roll Inflation,” JCAP 1901(2019) 040 [arXiv:1811.02175 [hep-th]].

[29] S. Passaglia, W. Hu and H. Motohashi, “Primordial blackholes and local non-Gaussianity in canonical inflation,”Phys. Rev. D 99 (2019) no.4, 043536 [arXiv:1812.08243[astro-ph.CO]].

[30] J. Garcia-Bellido, “Primordial Black Holes,” PoS EDSU2018 (2018) 042.

[31] G. I. Rigopoulos, E. P. S. Shellard and B. J. W. vanTent, “Quantitative bispectra from multifield inflation,”Phys. Rev. D 76 (2007), 083512 [arXiv:astro-ph/0511041[astro-ph]].

[32] N. C. Tsamis and R. P. Woodard, “Stochastic quantumgravitational inflation,” Nucl. Phys. B 724 (2005) 295[gr-qc/0505115].

[33] F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vaccaand G. Venturi, “Generation of fluctuations dur-ing inflation: Comparison of stochastic and field-theoretic approaches,” Phys. Rev. D 79 (2009) 044007[arXiv:0808.1786 [hep-th]].

[34] F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vaccaand G. Venturi, “Stochastic growth of quantum fluctua-tions during slow-roll inflation,” Phys. Rev. D 82 (2010)064020 [arXiv:1003.1327 [hep-th]].

[35] B. Garbrecht, F. Gautier, G. Rigopoulos and Y. Zhu,“Feynman Diagrams for Stochastic Inflation and Quan-tum Field Theory in de Sitter Space,” Phys. Rev. D 91(2015) 063520 [arXiv:1412.4893 [hep-th]].

[36] B. Garbrecht, G. Rigopoulos and Y. Zhu, “Infraredcorrelations in de Sitter space: Field theoretic versusstochastic approach,” Phys. Rev. D 89 (2014) 063506[arXiv:1310.0367 [hep-th]].

[37] I. Moss and G. Rigopoulos, “Effective long wavelengthscalar dynamics in de Sitter,” JCAP 1705 (2017) 009[arXiv:1611.07589 [gr-qc]].

[38] H. Assadullahi, H. Firouzjahi, M. Noorbala, V. Venninand D. Wands, “Multiple Fields in Stochastic Inflation,”JCAP 1606 (2016) 043 [arXiv:1604.04502 [hep-th]].

[39] V. Vennin, H. Assadullahi, H. Firouzjahi, M. Noorbalaand D. Wands, “Critical Number of Fields in Stochas-tic Inflation,” Phys. Rev. Lett. 118 (2017) no.3, 031301[arXiv:1604.06017 [astro-ph.CO]].

[40] G. Rigopoulos and A. Wilkins, “Inflation is always semi-classical: Diffusion domination overproduces PrimordialBlack Holes,” [arXiv:2107.05317 [astro-ph.CO]].

[41] G. I. Rigopoulos and E. P. S. Shellard, “The separateuniverse approach and the evolution of nonlinear super-horizon cosmological perturbations,” Phys. Rev. D 68(2003) 123518 [astro-ph/0306620].

[42] P. Binetruy, E. Kiritsis, J. Mabillard, M. Pieroni andC. Rosset, “Universality classes for models of inflation,”JCAP 1504 (2015) 033 [arXiv:1407.0820 [astro-ph.CO]].

[43] A. R. Liddle, P. Parsons and J. D. Barrow, “Formalizingthe slow roll approximation in inflation,” Phys. Rev. D50 (1994) 7222 [astro-ph/9408015].

[44] F. Cicciarella, J. Mabillard and M. Pieroni, “New per-spectives on constant-roll inflation,” JCAP 1801 (2018)024 [arXiv:1709.03527 [astro-ph.CO]].

[45] P. Friedrich and T. Prokopec, “Entropy production ininflation from spectator loops,” Phys. Rev. D 100 (2019)no.8, 083505 [arXiv:1907.13564 [astro-ph.CO]].

[46] P. H. Damgaard and H. Huffel, “Stochastic Quantiza-tion,” Phys. Rept. 152 (1987) 227.

[47] Oeksendal, Bernt K. (2003), “Stochastic Differen-tial Equations: An Introduction with Applications,”Springer, Berlin. ISBN 3-540-04758-1. Ludwig Arnold,“Stochastic Differential Equations: Theory and Ap-plications,” ISBN-10: 9780471033592 ISBN-13: 978-0471033592, Wiley-Blackwell, New York (1974). I. I.Grihman and A. V. Skorokod, “Stochastic Differen-tial Equations,” Springer (1972). Crispin W. Gardiner,“Handbook of Stochastic Methods,” ISBN10 978-3-540-70712-7, ISBN13 978-3-642-08962-6, Springer-Verlag(2009).

[48] Nico G. van Kampen, “Stochastic processes in physicsand chemistry,” North Holland 1981, 3rd edn., 2007,ISBN 0-444-89349-0.

[49] A. A. Starobinsky and J. Yokoyama, “Equilibriumstate of a selfinteracting scalar field in the De Sit-

17

ter background,” Phys. Rev. D 50 (1994) 6357 [astro-ph/9407016].

[50] T. Prokopec, N. C. Tsamis and R. P. Woodard, “Stochas-tic Inflationary Scalar Electrodynamics,” Annals Phys.323 (2008) 1324 [arXiv:0707.0847 [gr-qc]].

[51] J. J. Sakurai and J. Napolitano, “Modern quantum Me-chanics,” 2nd edition, Cambridge University Press, 2017.

[52] Pauli, W., Wave Mechanics: Volume 5 of Pauli Lec-tures on Physics (Dover Books on Physics, 2000) ISBN0486414620.