Path Integral for Stochastic Inflation: Non-Perturbative Volume Weighting, Complex Histories, Initial Conditions and the End of Inflation
aa r X i v : . [ h e p - t h ] M a r Path Integral for Stochastic Inflation: Non-Perturbative Volume Weighting, ComplexHistories, Initial Conditions and the End of Inflation
Steven Gratton ∗ Kavli Institute for Cosmology and Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, UK (Dated: March 11, 2010)In this paper we present a path integral formulation of stochastic inflation, in which volumeweighting can easily be implemented. With an in-depth study of inflation in a quartic potential, weinvestigate how the inflaton evolves and how inflation typically ends both with and without volumeweighting. Perhaps unexpectedly, complex histories sometimes emerge with volume weighting. Thereward for this excursion into the complex plane is an insight into how volume-weighted inflationboth loses memory of initial conditions and ends via slow-roll. The slow-roll end of inflation mitigatescertain “Youngness Paradox”-type criticisms of the volume-weighted paradigm. Thus it is perhapstime to rehabilitate proper time volume weighting as a viable measure for answering at least someinteresting cosmological questions.
I. INTRODUCTION
Inflation driven by the potential energy of some effec-tive scalar field [1–3] has become a common explanationof the starting state of the radiation-dominated hot bigbang model. A key reason for its acceptance is that smallquantum fluctuations during the last 60 or so efolds ofinflation can develop into almost scale invariant curva-ture perturbations [4–6] like those that we see today inthe cosmic microwave background fluctuations [7]. Cou-plings in the inflaton’s potential have to be chosen to bevery small in order to get the amplitude of the fluctu-ations suitably low. However, fluctuations in the scalarfield increase as the background energy density increases,so in certain circumstances the fluctuations might have asignificant effect on the evolution of large patches, lead-ing to “stochastic inflation” [8]. Such fluctuations mightlead to a situation in which part or even in some sensethe majority of the universe continues to inflate for alltime, i.e. “chaotic eternal inflation” [9].The advent of the “string landscape” [10, 11] withits complicated vacuum structure has reinvigorated thesearch for a suitable measure on inflationary histories insituations where more than one possible history can beconceived of. Much of the debate revolves around theextent to which predictions should be conditioned on ob-servations and, if more inflation leads to more observers,whether and how any “volume-weighting” should be im-plemented. For technical reasons much of this recentwork has focussed on models where the inflaton is ex-pected to “tunnel” from one vacuum state to anothervia bubble nucleation [12–20]. [21] is an exception, con-sidering random initial conditions in random potentials,and the “reheating-volume” approach has been appliedto both stochastic and bubble nucleation models [22–24].Quantum cosmological studies [25–34] provide comple-mentary perspectives. ∗ Electronic address: [email protected]
The approach discussed in this paper illuminates andexpands the approach to stochastic inflation and volume-weighting presented in [35], in which one follows the evo-lution of the inflaton in a λφ potential in proper timewith a Langevin noise term approximating the quantumfluctuations. There expectation values were calculatedfor the field history and perturbatively corrected for theeffects of volume weighting. By allowing for final-timeconstraints on the field value and considering weightingfield values at some time by either the volume at thattime or the volume at the final time, [35] began to directlyattack the two issues in the debate mentioned above. Thecurrent paper addresses more general inflationary modelsthan λφ and in some sense corrects the perturbative con-clusions of [35] via a non-perturbative treatment of vol-ume weighting by way of a path integral. The change inviewpoint is similar to that in going from the Heisenbergapproach to the Feynman approach in quantum mechan-ics when trying to address a question about the historyof the system. An early approach to a Langevin model ofstochastic inflation was presented by Hodges in [36]; morerecent work includes [37, 38]. Refs. [39, 40] also addresseternal inflation in a related manner. In contrast, muchof the early work on stochastic eternal inflation [41–45]attempted to follow in time the evolution of a probabilitydistribution for the inflaton with a Fokker-Planck equa-tion (analogous to the Schrodinger approach to quantummechanics). Such approaches typically broke down aftera finite time, when the probability became unnormaliz-able rising rapidly with field value, leading to the suspi-cion that Planck-scale effects might be vital in controllingthe theory and restoring predictivity. This led in part toproper time volume weighting falling out of favour as ameasure on eternal inflation. In addition, puzzles suchas the “Youngness Paradox” [46] (—if a fraction moreinflation produces exponentially more volume, aren’t themost common observers at a given time the youngestones conceivable?—) seem particularly acute with propertime volume weighting. We will see the surprising way inwhich a constrained path integral approach mitigates allthese issues and so it may be suggested that proper timevolume weighting should be reinstalled as a useful mea-sure for at least some calculations in stochatic inflation.This paper is organised as follows. First, a measure onslow-roll inflationary histories is presented. Saddle pointsof histories are discussed, and then volume-weighting isintroduced. The λφ model is studied in depth, and theway inflation typically ends is investigated. Finally thereis a discussion and conclusions. II. A MEASURE ON SLOW-ROLL HISTORIES
In this section we derive a measure on slow-roll infla-tionary histories, starting from the appropriate Langevinequation for slow-roll inflation (see e.g. [35]; note thatfor comparison with other works a 3 / (2 π ) factor omittedthere has been restored here):˙ φ + V ,φ H = 12 π H / n ( t ) (1)where n ( t ) is a Gaussian-normalized white noise termand H = H ( φ ) = p V ( φ ) / (3 M ) , (2)with M being the reduced Planck mass (henceforth as-sumed to be unity). One might think of Eq. (1) as de-scribing the evolution of a member of an ensemble ofphysical-Hubble-volume sized regions forward in time.Note that (2) determines H as a function of φ , and thusthe scale factor a at a time t as a function of the historyof φ up to time t .Now consider an arbitrary history φ ( t ). After a shorttime ∆ t , the field will be at φ ( t ) + ˙ φ ( t )∆ t , a changeof ∆ φ = ˙ φ ( t )∆ t . From (1), the change of the fieldshould be centered on − V ,φ ∆ t/ (3 H ) with a variance of H ∆ t/ (4 π ). So the probability of this segment of his-tory occurring is: r πH ∆ t e − π (cid:16) ˙ φ + V,φ H (cid:17) ∆ t/H . (3)Multiplying to obtain the joint probability for the entirehistory, and taking the limit ∆ t →
0, we obtain: P [ φ ( t )] Dφ ∼ e − R T L ( φ ) dt Dφ (4)with a “Euclidean Lagrangian” L = 2 π (cid:16) ˙ φ + V ,φ H (cid:17) H . (5)Furthermore, inspection of (3) suggests a change of vari-able that both renders the prefactor in (4) independentof field and makes the kinetic term in (5) canonical (upto a surface term): q ≡ Z πH / dφ (6) leading to P [ q ( t )] Dq = e − R T L ( q ) dt Dq (7)up to a numerical factor. Here L ( q ) = 12 ( ˙ q + f ( q )) (8)with f ≡ πV ,φ H / (9)expressed in terms of q . III. THE PATH INTEGRAL AND ITS SADDLEPOINTS
Given a general measure e − S [ q ( t )] Dq on histories q ( t )along with a specification of the class of histories to in-tegrate over, one may calculate the expectation value ofsome quantity of interest, A say, with a path integral: h A i = R Dq A e − S [ q ( t )] R Dq e − S [ q ( t )] . (10)Note that A can be of a very general nature, either localor non-local in time for example.As in the Feynman path integral approach to quantummechanics, it is often useful to look for saddle points inapproximating (10). Corresponding to (7) for examplewe would take S [ q ( t )] = S [ q ( t )] ≡ R T L ( q ) dt , and, cer-tainly in this case, finding saddle points is very simple,since the lagrangian is equivalent to that for a point massmoving in some potential. There even exists a conservedenergy. Furthermore, we can work with either q or φ , thesaddle points in either variable being equivalent.Once we have the saddle point solution for given initialand final data, we can use it to approximate the proba-bility density for that final data given the initial data byintegrating (7) in a gaussian approximation. The leadingterm is simply the exponential of minus the action eval-uated for the saddle point. By varying the field value atthe final time T and recalculating the saddle point solu-tion and the (approximate) probability density, we canbuild up a picture of the probability distribution functionat the final time. By repeating the procedure for differentfinal times, we can build up a picture of the evolution ofthe probability distribution function in time. For (7), we An alternate derivation following the lines of [47] can yield adeterminant correction to the measure coming from the changeof variables from noise realizations to field realizations. For the q variable for a λφ potential, this determinant is independentof field so both derivations agree precisely. Given the alreadyheuristic nature of our starting point, Eq. (1), we do not considersuch corrections further in this work. thus find an approximate solution to the Fokker-Planckequation corresponding to (1).We can also calculate the expectation value of A inthe saddle point approximation. For a fixed final fieldvalue, we just evaluate the quantity for the related saddlepoint solution. To relax the final field value constraint,we compute the average of A calculated for all final fieldvalues, weighted by the exponential of minus the action(possibly multiplied by the gaussian prefactor for higheraccuracy) for the corresponding final field value.Let us quickly derive some useful results. Assumingthe action S is expressible as an integral over a local-in-time Lagrangian L , then we can introduce a momentum p q ≡ ∂L∂ ˙ q (11)and Hamiltonian H ′ = p q ˙ q − L (primed to avoid confu-sion with the Hubble parameter H ) and correspondingenergy E . Just as in classical mechanics (see e.g. [48]),evaluating the action as a function of end point and endtime, we have δS = p q δq − H ′ δt . So, at a fixed final time T , the action is extremized for a solution which bothobeys the equations of motion and has p q ( T ) = 0. This,neglecting prefactor corrections, gives the extremum inthe probability distribution function for q at time T . As long as L − L does not involve ˙ q , p q ( T ) = 0 implies˙ φ = − V ,φ / H at time T , i.e. a trajectory correspondingto an extremum in the probability distribution at time T is slow-rolling as T is approached. This does not nec-essarily imply that the trajectory slow-rolls all the wayfrom 0 to T , nor that extrema of the probability distri-bution evolve according to slow-roll as T is varied. How-ever, in the simple case when L is just L , both of thesestatements do in fact hold. A helpful way to verify suchstatements is to look at the (conserved) energy associatedwith each saddle point solution. For L , this is just E = 12 ˙ q − q (12)and for a solution that slow-rolls at the end time T , wesee E = 0. As E is conserved along the saddle point, wesee that ˙ q must equal q back along the entire trajectoryto q = q ( φ ) at t = 0, i.e. the saddle point is just theslow-roll solution evolved from the initial condition tothe time T . Hence as T changes the position of the peakalso follows the slow-roll solution in this case. Note that here, unlike in most classical mechanics applications,one does have to be careful not to discard total integrals, which,while not altering the saddle point solution, affect the total valuefor the action and so the probability for a history.
IV. THE VOLUME-WEIGHTED PATHINTEGRAL
Now we volume-weight. We do this by “reweighting”the e − S [ q ( t )] Dq measure from above by an appropriateterm, and then renormalizing. We might imagine thephysical-Hubble-volume sized regions followed above asbeing “probes” of much larger volumes of space that areinflating and so expanding in time. Thus the term istypically just the “final” volume a ( T ) = e R T H ( q ( t )) dt . (13)Because of the local-in-time nature of the exponent, wesee that such volume weighting can be incorporated verysimply by thinking of S as an integral over t of a moregeneral “Euclidean Lagrangian” L V = L − H. (14)We have only had to add a local-in-time term to thepotential for q . This extra term alters the constrainedhistories relative to the non-volume-weighted ones withthe same boundary conditions.The volume-weighting term, only involving the fieldand not its derivative, will not affect the expression forthe momentum in terms of the field and its derivative. So,as for the non-volume-weighted case discussed above, themost probable trajectory will obey the slow-roll conditionat the very end.However, unlike before, this trajectory will not haveslow-rolled all the way from the initial condition; thereare two additional effects that cannot generally cancel.First, the equation of motion now has an extra field-dependent term. Second, the expression for the energy E has an additional +3 H ( φ ) term, so, evaluating this at theend of the trajectory, the energy of the solution is movedfrom zero to E = 3 H ( φ ( T )). Thinking momentarily of T as a function of q at time T for the most probablesolution, we have T = Z qq (0) dq ′ p E ( q ) − V ( q ′ )) . (15)Changing E and V as discussed to incorporate volume-weighting will change T ( q ) and so q considered as a func-tion of T : the peak of the probability distribution func-tion does not now follow slow-roll.We can now ask whether or not inflation ends in therolling-volume-weighted average. If it does, the peak ofthe probability distribution function should pass througha field value corresponding to a small value of H , i.e. theconserved energy should be able to have a small value. One could also for example consider weighting by the vol-ume at an intermediate time for calculating a final-field-value-constrained “rolling” volume average.
Evaluating the energy, expressed in terms of φ , at theinitial condition thus leads to the condition:2 π ˙ φ (0) H − π H V ,φ H + 3 H ≈ φ , if the potential term contributionto the left hand side is positive then inflation cannot end.So if H > π V ,φ
27 (17)then inflation cannot end in the volume-weighted aver-age (c.f.[39]). This precise constraint is consistent withthe qualitative arguments of e.g. Guth [46] comparing theclassical movement of the field to the quantum fluctua-tions in the field over a Hubble time. We may say thatinflation is indeed “eternal” if the field starts at a valuesuch that (17) is satisfied. Note that for λφ this requires φ > (32 π /λ ) / .Let us assume that the field starts above the eter-nal inflation threshold and ask what happens. Does thevolume-weighted field asymptote to some constant value,and if so can this value be above or below the startingvalue? Or does the field average run away to a place of in-finite energy density, in either finite or infinite time? Weare minded here of the early results looking at volume-weighted eternal inflation by solving the Fokker-Planckequation for say a λφ potential; there it seemed thatthe probability distribution lost its extrema after a finiteamount of time, becoming unnormalizably peaked at aninfinite field value. This led to the imposition of arbi-trary boundary conditions at the Planck density and theview that the volume-weighted field would quickly tendto its largest possible value and stay there inflating atthe maximum possible rate. We indeed find, investigat-ing λφ as a specific example as discussed in the follow-ing section, that the average can stop existing after afinite time. Above we showed that the paths correspond-ing to extrema of the probability distribution must endin slow-roll. This applies to both maxima and minima,and a minimum in the probability distribution will de-limit a formal unbounded rise in probability towards veryhigh energy from a physical region of field values withits own maximum. As time goes on, the maximum andminimum merge; the probability distribution steadily in-creases with field energy. To see whether trajectorieswith Planckian energy densities are important or not forthe disappearance of the average, we now though canlook at the critical saddle point history and see whetheror not it approaches the Planck density at any stage. Forsmall coupling it turns out it does not and so we concludethat Planck scale effects will not affect the disappearanceof volume-weighted average.One may be concerned that the disappearance of theaverage is indicating a failure with the whole approach.However, we can continue to find constrained paths tolower field values, at least for a finite window of larger time intervals. So perhaps the correct interpretation issimply that answering the question “what is the fieldaverage on a surface of constant proper time?” is becom-ing problematical. But because we can still answer otherquestions, about constrained paths say, proper time vol-ume weighting itself is not obviously failing at this stage.Pushing to larger time intervals still, we find for λφ at least that real histories connecting the inflationary ini-tial conditions to low field values cease to exist. Surelyeven constrained proper time volume weighting is failingnow? This is not necessarily the case because complex histories now emerge that link the initial and final condi-tions. Further, as we will see below, these complex pathsare very close to being real towards the end of inflation,and indeed basically become “classical” slow-roll trajec-tories, insensitive to the initial field value or indeed thetime interval between the initial and final conditions. V. λφ IN DEPTH: COMPLEX HISTORIES,INITIAL CONDITIONS AND THE END OFINFLATION
For λφ , q is proportional to 1 /φ , which is in turnproportional to the Hubble radius which we denote hereby R and work with in order to allow for easy comparisonwith [35]. Introducing the (dimensionless) constants α =8 p λ/ β = p λ/ /π , we find L V = (cid:16) ˙ R − αR (cid:17) β − R . (18)Saddle-point histories satisfy¨ R = α R + 3 β R (19)with a conserved energy E = 12 β ˙ R − α β R + 3 R . (20)See Fig. 1 for the associated effective potential that the R variable feels. (Note that in all plots R and t have beenrescaled to absorb α and β via t → αt , R → ( α/β ) (2 / R .)As discussed in Sec. IV, the momentum p R = 1 β (cid:16) ˙ R − αR (cid:17) (21)has to be zero at the end of a history that finishes atan extremum of the probability distribution function attime T . Hence, from (20), the energy of such a path is3 /R ( T ), which is small for weak coupling (of order λ − / )if R ( T ) corresponds to φ ∼ R ever becomes less than p β /α (the zero of the effectivepotential in Fig. 1). - - _ eff FIG. 1: Plot of the effective potential that the R variablemoves in. Fig. 1 is very helpful for understanding the nature ofthe (real) saddle point solutions, illustrating the discus-sion of Sec. IV. Let us look for solutions connecting R = r i to R = r f (“i” for initial, “f” for final). If r f > r i ,there are two classes of solutions differentiated by thesign of the initial velocity. One rolls up the potential,turns round and rolls back down past r i on the way to r f ; the other rolls straight down the potential from r i to r f . If r f < r i , there are again two classes of solution, nowdifferentiated by the sign of the final velocity. In one, R rolls straight up the potential from r i to r f , in the other, R rolls up the potential passing through r f , then rollsback down to r f . If r f = r i , there is only one class ofsolution, R rolling up the potential and then back down.In all cases, varying the initial speed, or equivalently theenergy E , changes the time T needed to go from r i to r f .Scanning over r f , E and the velocity sign and recordingthe time T each solution takes gives us complete informa-tion about the behaviour of the probability distributionfunction for R as a function of T . Note though thatthere is no guarantee that arbitrarily large values of T will be obtained in the scan, and indeed T turns out tobe bounded when volume weighting is switched on.If we temporarily focus on the subset of histories withzero final momentum (and so with energy E = 3 /r f withvolume-weighting, or with E = 0 without), correspond-ing to the extrema of the probability distribution func-tion, we can build up a sketch of the loci of the maximaand minima of the probability distribution function as afunction of time as in Fig. 2. Without volume-weighting,a solution exists for any T (the field spending longerand longer at small R as T increases), with r f alwaysgreater than r i , and the distribution moves (exponen-tially in time) to larger R as time passes. Switchingon volume weighting corresponds to adding a repulsiveforce, requiring R to start with a more negative velocitythan for the non-weighted case with the same r f and T .Thus R gets smaller more rapidly as expected; volume-weighting favours higher field values. Starting well belowthe eternal inflation threshold, the picture is qualitativelysimilar to the non-volume-weighted case. Starting above R t r_0 maxmin R t r_0 maxmin FIG. 2: Sketch of the loci of the minima and maxima ofthe volume-weighted probability distribution function for R as a function of time. The left panel is for starting below theeternal inflation threshold, while the right panel is for start-ing above the eternal inflation threshold. (Without volume-weighting the plot would be qualitatively similar to the “max”branch of the left hand panel.) the eternal inflation threshold, the picture changes sig-nificantly however. The steep, “brick-wall”, nature ofthe repulse volume-weighting term in the effective po-tential means that there is in fact an upper limit on howmuch T can be increased by increasing the initial speedof R . After this time there are no extrema; the proba-bility density increases monotonically towards small R .We are able to conclude that at some intermediate time,at some R > r i , the peak of the probability distributionturns around; the inflaton begins to climb back up its po-tential in the volume-weighted average. The largest valueof r f attained can be deduced by equating the effectivepotential at R = r i to the energy E = 3 /r f evaluated atthe end.Returning to the general case, it may seem strange thatconstrained saddle point solutions linking r i to r f fail toexist for too large of a time difference. As mentionedabove, the paradox is resolved by realizing that thereis no necessity for the saddle point histories to be real.Just as in contour integration, where one may deform realcontours into the complex plane to pass through a saddlepoint in order to apply the method of steepest descents,we can do likewise here. Indeed, the use of complex his-tories has a precedent in the Euclidean “No-Bounday”approach to quantum cosmology, where for “large” fi-nal three-geometries the Euclidean path integral has acomplex saddle point with a Lorentzian part [25]. Weneed only preserve our boundary conditions, namely that R (0) = r i and that R ( T ) = r f . We see straight away thatthe imaginary component of R has to be zero at bothends, but that there is no such constraint on the imagi-nary component of ˙ R at the ends. Decomposing (19) intoreal and imaginary parts, we can visualize R as a pointmoving in a two-dimensional force field as heuristicallyplotted in Fig. 3. We note the (unstable) zero-force loca-tions specified by R = − β /α . By tuning the solutionso that it approaches one of these points with near zero - - - - H R L I m H R L FIG. 3: Heuristic plot of the force field determining the mo-tion of R in the complex R -plane. Note the zero-force “loi-tering points” at the solutions of R = −
3. Planckian energydensity, λφ ∼
1, corresponds to the rescaled R shown beingsmall, of order λ / for weak coupling. An eternal-inflationarystarting point would be along the real axis between 0 and √ φ ∼
1, corresponds to the rescaled R shown becoming large and positive, of order λ − / . speed, it is possible for R to “loiter” there for as long as isneeded. Solutions with a long loitering period must havea complex energy very close to that of the effective po-tential evaluated at the loitering point in question. Thiscomplex energy determines solutions that asymptoticallyreach the loitering point in the future from r i or from thepast from r f . Appropriate deviations in the initial veloc-ities will “connect up” the two asymptotic solutions andmake the total solution last for the desired finite time T .For solutions linking inflaton values corresponding tostarting above the eternal-inflationary threshold to fieldvalues towards the end of conventional inflation, the ap-propriate loitering points are the ones with a positivereal component and non-zero imaginary component of R . These will provide a conjugate pair of histories. Eachhistory will approach its respective loitering point andthen roll back towards the real axis out to large positivevalues of R .Focusing on a single member of a conjugate pair forclarity, the way the solution reaches r f will become al-most independent of how large T is, as long as T is suf-ficiently large. We thus see in a precise way how eternalinflation “loses memory” of initial conditions, in that, atsufficiently late times, the way inflation typically ends isvery insensitive to the initial field value.Note that for weak coupling the history need not goparticularly close to R < ∼ t Φ FIG. 4: Plot of the loitering solution (solid line) departingfrom the slow-roll solution (dashed line) to the past of a regionwhere inflation ends. The difference only becomes significantwhen the field approaches the eternal inflationary regime; theend of inflation is classical. points corresponds to φ ∼ λ − / , of order the eternalinflation threshold. Hence conclusions drawn from thehistory may hoped to be insensitive to any Planck-scalecorrections to the model.The imaginary part of a trajectory corresponding tothe late-time end of eternal inflation for weak coupling isvery small, and the real part of the field basically slow-rolls, as illustrated in Fig. 4. These statements can bemade quantitative by rearranging (20) to express ˙ R interms of R and E ,˙ R = αR s E − β R α R , (22)the fractional correction over the slow-roll result ( ˙ R = αR ) being small when the modulus of R is well awayfrom the eternal inflationary regime.We have just seen that where stochastic eternal in-flation ends, it basically ends classically. This mightappear counter-intuitive, given the Youngness Paradoxarguments about proper-time volume weighting. Con-sider looking for regions where the inflaton, if it slow-rolled, would have either one or two efolds say left togo. The Youngness Paradox would suggest that thereare exponentially many more regions with one efold leftthan regions with two efolds left, the latter histories be-ing “younger” and so having had more inflation in theirpast. One might have also thought that to the past ofany of these regions the field would be much higher up itspotential than slow-roll would suggest, perhaps even upat Planck-density values, since such histories would haveexponentially more volume. Our result is not inconsis-tent with the first conclusion but suggests that volume-weighting does not sufficiently dominate over classicalmotion for the second conclusion to apply also. Thusstandard calculations of inflationary density perturba-tions are probably safe even in eternal inflation as longas the coupling is weak. VI. DISCUSSION AND CONCLUSIONS
We can assemble what we have learned above intoa somewhat cogent picture of volume-weighted stochas-tic eternal inflation. The field must start off above theeternal inflationary threshold, and then we soon see thevolume-averaged field stop decreasing and turn aroundand begin increasing, indicating that volume effects areoutweighing classical drift. From this we may hope thata late-time “steady state” situation will arise with late-time results dominating any averaging. After a finiteproper time, the volume-averaged field ceases to exist;the system is dominated by strong fluctuations and a“global” picture breaks down. Nontheless, we may chooseto focus on the observationally relevant but rare regionsof the universe where inflation happens to end. Then wefind that inflation ends in practically the same slow-rollmanner on all proper time slices and hence some levelof predictability is restored. The saddle point historiesdeviate into the complex plane rather than continue tovalues far above the eternal inflation threshold, indicatingthat the conventional view of the inflaton as jumping upand down on its potential in the eternal inflation regimemight be too simplistic. Because when inflation does endit basically ends in slow-roll, conventional density per-turbation calculations should still apply, preserving thesuccessful predictions of conventional treatments of infla-tion.The general techniques and insights of this papershould apply to many large-field models of inflation. Itwould be interesting to investigate potentials with mul-tiple vacua. Indeed, for “Mexican-hat” type potentials, V = λ ( φ − φ ) , one can analytically obtain an expres-sion for q in terms of φ and so obtain an explicit formulafor the effective potential for q . Thus one could investi-gate volume-weighting for small-field models of inflationwhere would might expect its effect to be less pronouncedthan for the large-field case studied here. (Note that anearly work [49] discusses an approximate path integraltreatment of the behaviour of the inflaton in a “new” in-flationary potential.) Numerical investigation of the de-terminant prefactor would be helpful in getting an idea inhow “classical” the histories really are. The author haschecked numerically that there are no negative modessatisfying the relevant boundary conditions for a sampleof representative (real) histories, as one would hope. Itwould also be possible to go beyond slow-roll, obtain-ing fourth-order equations for the saddle point histories,though the precise way in which the quantum fluctua- tions are modelled might need to be thought throughmore carefully.As in quantum mechanics, we have seen that a pathintegral approach is particularly useful when asking time-dependent questions and looking for semi-classical his-tories. It has given us a technique for calculating involume-weighted eternal inflation that is relevant for ob-servations. We have been able to demonstrate how infla-tion typically ends normally even with volume-weightingin a manner insensitive to the precise initial conditions.By retreating from demanding a global picture of theuniverse at all times and rather adopting a more “top-down” observationally relevant approach [28, 29, 35] thepath integral has allowed us to push much further thanin the Fokker-Planck approach without having to worryabout Planck density issues. We have also obtained a dif-ferent result about the behaviour of the volume-weightedaverage than in the Langevin treatment of [35]. This ispossibly because that work only perturbatively expandedaround the classical solution, implicitly forcing one toconsider only the subset of histories in which inflationhas to end.Finally, let us return to the question of proper-time vol-ume weighting itself. Rather than any intrinsic flaw inthe scheme, perhaps it was the gauge-dependence of thequestions that proper time volume weighting encouragedone to ask that led to the weighting getting a bad rep-utation (a canonical example of such a gauge-dependentquestion being “which value of the inflaton is most likelyat a given time?”). A question that we have addressedin this paper is “how does inflation end at a given propertime?”. Seeing that the answer only depends very weaklyon what that time actually is, we have been able to ob-tain a satisfactory answer to the more general reasonablequestion “how does inflation end?” even using proper-time volume weighting. So for at least some physicallyrelevant questions perhaps proper-time volume weightingis not so bad after all. Acknowledgments
We thank Anthony Aguirre, Tom Banks, Paolo Crem-inelli, Jaume Garriga, Alan Guth, Thomas Hertog,Antony Lewis, James Martin, Paul Steinhardt, NeilTurok and Toby Wiseman for helpful discussions. Thisresearch was supported in part by a minigrant from TheFoundational Questions Institute (fqxi.org). I am sup-ported by STFC. [1] A. H. Guth, Phys. Rev.
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