TThe Stochastic Axion Scenario
Peter W. Graham and Adam Scherlis Stanford Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305
For the minimal QCD axion model it is generally believed that overproduction of dark matterconstrains the axion mass to be above a certain threshold, or at least that the initial misalignmentangle must be tuned if the mass is below that threshold. We demonstrate that this is incorrect.During inflation the axion tends toward an equilibrium, assuming the Hubble scale is low andinflation lasts sufficiently long. This means the minimal QCD axion can naturally give the observeddark matter abundance in the entire lower part of the mass range, down to masses ∼ − eV (or f a up to almost the Planck scale). The axion abundance is generated by quantum fluctuations ofthe field during inflation. This mechanism generates cold dark matter with negligible isocurvatureperturbations. In addition to the QCD axion, this mechanism can also generate a cosmologicalabundance of axion-like particles and other light fields. Contents
I. Introduction II. Summary III. Dynamics of Axions During Inflation
IV. Sub-Horizon Modes and Isocurvature Bounds V. Results
VI. Inflationary Sector H I VII. Conclusions Acknowledgements A. Misalignment Angle and PQ Scale f a B. Fokker-Planck Formalism and Inflationary Backreaction C. Classical Rolling Constraint References a r X i v : . [ h e p - ph ] A ug I. INTRODUCTION
The identity of dark matter is one of the major outstanding questions in physics. The mass of the dark matterparticle is enormously unconstrained, ranging from 10 − eV for fuzzy dark matter models to the Planck scale ≈ eV, or even higher for dark matter composed of primordial black holes.The QCD axion is a popular and well-motivated candidate for light ( m a (cid:28) eV) dark matter. In the “post-inflationary” scenario for axion dark matter, the population of axions is produced after inflation, when Peccei-Quinnsymmetry breaks spontaneously. This scenario predicts a unique value for the axion mass in the µ eV range (or axiondecay constant f a ∼ GeV). However, if PQ symmetry is already broken during inflation, the axion mass can bein a broad range roughly 10 − − − eV or even beyond and still be dark matter. With a relatively short durationfor inflation, the axion abundance is set by the initial average value of the axion field, f a θ (where θ is the initialmisalignment angle). In order for the axion to be much lighter than the post-inflationary value, θ (cid:28) f a ) QCD axion canin fact naturally have the correct dark matter abundance without tuning or anthropics (see e.g. [5–14]). These modelsgenerically have new particles coupling to the axion or the Standard Model.The question of whether axion dark matter can naturally have a light mass or high f a is an important one. Theaxion is an extremely well-motivated dark matter candidate and significant effort is being expended to search for it(see e.g. [15–22] and many others). The axion mass is critical to determining the type of experimental search used todetect it. Any guidance from theory on the axion mass is therefore important. Lighter axions arise from PQ-breakingscales around fundamental scales such as the grand unified theory (GUT), string, or Planck scale and are thereforetheoretically very well motivated (see e.g. [23–27]). There are in fact several experiments aiming to detect axionswith low masses. In particular CASPEr [28–31] aims to push down to the QCD axion at the lowest masses, f a around the GUT to Planck scales. Additionally, electromagnetic LC-circuit experiments, e.g. LC Circuit [32], DMRadio [33, 34], or ABRACADABRA [35], aim to push to QCD axion masses somewhat below cavity searches in the“post-inflationary” region. Hopefully the combination of all axion experiments will be able to cover the entire allowedQCD axion mass range. II. SUMMARY
In this paper we observe that even the simplest QCD axion model can have the correct dark matter abundance atlow axion mass, new particles are not required. All that is necessary is a long period of low-scale inflation. Duringa long enough period of inflation, the axion is naturally driven to an equilibrium distribution of field values. Thisequilibrium can produce the correct dark matter abundance under one constraint on the parameters: the Hubble scaleof inflation H I and the mass of the axion m a . We show our results for this region in Fig. 2.For long durations of inflation, if the Hubble scale H I is lower than the QCD scale, θ relaxes to a small value [37].The axion abundance is naturally suppressed because the axion is classically driven to the minimum of its potentialduring inflation. However the axion abundance cannot go to zero because of quantum fluctuations. A nonzeroabundance is generated by quantum fluctuations of the axion field during inflation. The competition between theclassical force driving the axion field towards zero and the stochastic quantum fluctuations which can drive the axionfield up its potential results in an equilibrium. The axion field is naturally driven towards this equilibrium duringinflation. Roughly the quantum fluctuations of the axion can be viewed as arising from the de Sitter temperatureduring inflation. The axion abundance can be estimated from the “thermal” equilibrium value of the axion field duringinflation, as discussed in greater detail below. This makes the initial θ after inflation naturally small but nonzero.And so, in particular, if H I is in the correct range, the axion naturally gives the measured dark matter abundancefor axion masses as small as ∼ − eV. III. DYNAMICS OF AXIONS DURING INFLATION
Inflationary fluctuations of spectator fields are usually considered in the context of modes whose wavelengths arewithin the Hubble horizon today – that is, they are visible in the power spectrum of a field. However, if inflation lastslonger than 60 e-folds, it also produces modes which never re-enter the horizon. These modes are observable because Note that such a period of long, low-scale inflation is also used by many relaxion [36] solutions to the hierarchy problem. they contribute to the average value of any spectator field over our Hubble patch (see e.g. [38–45]). In particular, forthe QCD axion, this average value (in the early universe) is the misalignment angle responsible for post-inflationaryaxion dark matter. If inflation lasts long enough, the inflationary contribution dominates and the Hubble scale ofinflation determines the misalignment angle and axion abundance. The distribution of misalignment angles for the“unrolled” axion field is multimodal and continues to evolve indefinitely [46].During inflation, the average values of scalar fields fluctuate around the minimum of the field’s potential V . For lightfields, which are overdamped by Hubble friction, these fluctuations can be quite large if inflation lasts long enough forthem to accumulate. A free scalar ( V = m φ ) will typically end up with energy density of order V ≈ T dS , where T dS = H I / π is the de Sitter temperature; the field will therefore be displaced by φ ≈ H I /m from its minimum.As inflation progresses, modes of the field are stretched until their wavelengths are longer than the Hubble radius,after which they are mostly “frozen”. Each mode has amplitude of order H I , and gives a random “kick” of this sizeto the average field value as it crosses the horizon, producing random-walk behavior for the field value. The field alsoslow-rolls down the potential towards its minimum (if it is massive with m < H I ); without the quantum fluctuations,this would relax the field to a very small value. These two opposing tendencies are in equilibrium when the averagefield value is of order H I /m ; while the average motion of the field value is always towards zero, the random fluctuationsprevent it from settling exactly there.The QCD axion is massive as long as T (cid:46) Λ QCD , with mass m a ≈ Λ QCD /f a and CP-violating angle θ = φ/f a .Therefore, if H I is slightly below Λ QCD , inflationary fluctuations will produce a misalignment angle of order θ ≈ H I /m a f a ≈ H I / Λ QCD . The misalignment required for axion dark matter is between 10 − and O (1) depending on f a .QCD axion dark matter can easily be obtained from these fluctuations for any f a between 10 GeV and the Planckscale, depending on H I . Because the fluctuations are homogeneous over super-Hubble scales, the cosmic variance in θ is high and the relationship between f a and H I is approximate.A similar mechanism to ours has been considered for vector dark matter [47]; however, for vectors, sub-horizonmodes (generated in the last 60 or less e-folds) dominate, the long-wavelength modes considered in this paper arenegligible, and the vector dark matter abundance has low cosmic variance. A. Setup
Consider the average value of a scalar field (with m (cid:28) H I ) over one Hubble patch. Throughout this paper, we willrefer to this simply as φ . This value is approximately given by the sum of long-wavelength modes with k (cid:28) aH I ,which are homogeneous on the scale of a Hubble patch; shorter wavelengths k (cid:29) aH I are averaged over many periodsand contribute much less to φ . As the scalar field evolves during inflation, the k (cid:28) aH I modes are “frozen”, but decayslowly towards zero; equivalently, φ slow-rolls towards the minimum of its potential, overdamped by Hubble friction.Meanwhile, new modes with k ≈ aH I “leave the horizon” and become part of φ , with each order of magnitude in k contributing an amplitude of order H I . The phase is random, so these new modes produce a random-walk behaviorfor φ .Instead of focusing on the stochastically-evolving value φ in a specific Hubble patch, we will track the distribution ρ ( φ, t ) which describes the frequency of different values of φ across many patches. The fraction of patches withfield values in a small interval between φ and φ + d φ is given by ρ ( φ )d φ , and (cid:82) ρ d φ = 1. This distribution evolvesdeterministically. The potential for φ tends to concentrate the distribution near its minimum, while the randomcontributions cause ρ to diffuse and even out. These opposing effects determine the typical width of ρ when it reachesequilibrium.We assume that the potential V ( φ ) is negligible compared to V I , the energy density due to the inflaton; thisassumption holds up for the QCD axion as long as H I (cid:29) − eV. For more discussion of backreaction effects, seeSec. VI D and Appendix B. B. Massive Free Scalar
As a simple example, consider a free scalar field V = m φ and Gaussian initial condition ρ ( φ, ∝ exp( − ( φ − µ ) / σ ) . (1)In this case, the distribution remains Gaussian for all time.If there were no fluctuations, every point in the distribution would independently evolve according to the slow-rollequation, ˙ φ = V (cid:48) ( φ )3 H I = − m H I φ (2)This has the solution φ ( t ) ∝ exp( − ( m / H I ) t ). As each point φ follows this trajectory, the distribution concentratesnear zero. More precisely, the distribution ρ ( φ, t ) is uniformly rescaled by a factor of exp( − ( m / H I ) t ) horizontallyand exp(( m / H I ) t ) vertically. Therefore, the mean and standard deviation of the Gaussian would follow the sameevolution, ˙ µ | no diffusion = − m H I µ (3)˙ σ | no diffusion = − m H I σ (4)Instead of the standard deviation, it will be more useful to work in terms of the variance σ :˙( σ ) (cid:12)(cid:12)(cid:12) no diffusion = 2 σ ˙ σ = − m H I σ (5)Now consider the case of diffusion with the potential V switched off. The Gaussian spreads, σ ∼ √ t , so the varianceincreases linearly: ˙ µ | no rolling = 0 (6)˙( σ ) (cid:12)(cid:12)(cid:12) no rolling = H I π (7)With both effects present, we have ˙ µ = − m H I µ (8)˙( σ ) = H I π − m H I σ (9)The first term in ˙( σ ) is the diffusion (random walking) of the field, which tends to increase variance linearly; thesecond term is due to the quadratic potential squeezing the field closer to its minimum. These two effects cancel outat an equilibrium value of σ given by σ f = 3 H I π m (10)Note that this corresponds to the general formula for the equilibrium distribution given in Eq. (13). In terms of∆ = σ − σ f , we have ˙∆ = − m H I ∆ (11)Therefore, the mean and variance of the distribution both approach their equilibrium values exponentially at a rateof order m / H I . If the field begins at a single initial value M (Gaussian with µ (0) = M, σ (0) = 0), it will closelyapproximate the equilibrium distribution after a small multiple of H I m log( M/σ f ) . (12)Note that a time of order H I /m , during inflation, is N = H I /m e-folds.Because the Fokker-Planck equation for this system is linear, any initial distribution restricted to [ − M, M ] willreach equilibrium on the same timescale. More rigorously, we can find the full set of quasinormal modes of theFokker-Planck equation and show that the slowest decay rates are of order m / H I ; see Appendix B. C. Massless Compact Scalar
Now consider a scalar field φ = f θ with a compact field range, so that φ = 0 and φ = 2 πf are identified. In thiscase, there is no potential so ρ diffuses into a uniform distribution.Starting from a Dirac delta, the variance of the distribution for the field, f θ , will increase at a rate of H I / π . Thevariance of θ will therefore increase at a rate of H I / π f . The distribution will become uniform once this varianceis (cid:38)
1, which will take a time of order 4 π f /H I , so that N (cid:38) π f /H I .A more precise calculation (Appendix B) gives a relaxation time of 8 π f /H I e-folds for this case. The quasinormalmodes are sinusoidal in this case. D. General Scalar
The cases above are easy to work with and are good approximations to the QCD axion potential for H I (cid:28) Λ QCD and H I (cid:29) Λ QCD respectively, as we will see in Sec. V A.More generally, any initial probability distribution for the value of a scalar field will asymptotically approach anequilibrium distribution[38] (see Appendix B) which looks like a Boltzmann distribution with temperature of order H I : ρ f ( φ, t ) ∝ exp (cid:18) − π V ( φ )3 H I (cid:19) (13)This does not, however, mean that the field is thermalized; rather, it has a misalignment angle which gives itshomogeneous mode an energy density of order H I (unless the potential is very flat, as in the compact case above).This energy behaves as dark energy while H I (cid:29) m and becomes a condensate of cold matter at H I (cid:46) m . As wewill show, this matter quickly becomes nonrelativistic, contrary to what would be expected from a bath of radiationproduced at temperature H I .The distribution of field values is described by a Fokker-Planck equation (Appendix B) with a diffusion term (for thekicks) and a classical force term (for the classical slow-rolling). These two opposing tendencies bring the distributionto the above equilibrium, over some characteristic relaxation time. This relaxation time is the time required forinflationary super-horizon modes to dominate over whatever long-wavelength modes existed at the beginning ofinflation; in other words, the timescale at which sensitivity to initial conditions goes away. Note that the varianceof the distribution is entirely cosmic variance: each patch takes on one average value by definition, so we can onlyobserve one sample from this distribution. IV. SUB-HORIZON MODES AND ISOCURVATURE BOUNDS
The modes smaller than the horizon (after inflation) produce inhomogeneities in the field, which are observable asisocurvature perturbations. These modes are produced during the last 60 or less e-folds, so the mechanism we considerdoes not affect them. As H I becomes smaller, these perturbations become negligible relative to the total density;for the QCD axion, they are below current observational bounds for H I (cid:46) TeV and f a (cid:38) GeV. In particular,there is negligible isocurvature for H I (cid:46) Λ QCD , where the axion mass is nonzero. For f a (cid:46) GeV, if the QCDaxion makes up most of dark matter, the value of θ approaches π and anharmonic effects increase the isocurvatureperturbations [48, 49].For low H I , the dark matter produced by our mechanism during long periods of inflation is extremely cold; thisis due to the fact that all of the modes produced significantly prior to the last 60 e-folds have unobservably smallgradients. It is fairly straightforward to calculate the velocity of dark-matter axions at matter-radiation equality:the total energy density of dark matter is comparable to radiation, so it is parametrically T (there are only a fewrelativistic degrees of freedom at this point). Now, consider the kinetic energy density of all modes produced during ane-fold of inflation. When these modes re-enter, they have kinetic energy density ( k/a ) φ k ≈ H H I ≈ ( H I /M P ) T . Ifthey are relativistic ( k/a > m ), they will redshift as a − ∼ H (during the radiation-dominated era), so this formulawill hold for all relativistic modes at any point until matter-radiation equality. Therefore, the kinetic energy densityof the axion field is parametrically ( H I /M P ) T , disregarding a logarithm. The velocity of the axions is then v ≈ p /m ≈ H I /M P (14) v ≈ H I /M P (15)The value of H I /M P ranges from 10 − to 10 − for our parameter space.At much earlier times, when H > m a , the kinetic energy density of axions is still ( H I /M P ) T but the total densityis H I . This gives v ≈ H/H I ≈ ( T /T rh ) , where T rh is the reheating temperature. (This is as expected; kineticenergy of relativistic modes dilutes as a − but total energy density is constant at this stage.) Therefore, the axion isnonrelativistic immediately after reheating, and becomes cold very quickly. V. RESULTSA. QCD Axion
The QCD axion has a mass which depends on temperature, which we take to be T dS = H I / π . However, thismass has a constant value of m a ∼ Λ QCD /f a for H I (cid:46) Λ QCD , where Λ
QCD is the QCD scale. To be more precise,however, we should disambiguate two definitions of the QCD scale. One is based on the maximum height of the axionpotential, which is χ (0) ≈ (75 MeV) [50]; this is the topological susceptibility of QCD, χ ( T ), at zero temperature.The other definition is T c ≈
130 MeV; this is the temperature below which the axion mass is constant [50], near theQCD phase transition.Incorporating this distinction, we find that m a = (cid:112) χ (0) /f a ≈ (75 MeV) /f a when H I (cid:46) πT c ≈
800 MeV. Thisorder-of-magnitude separation in scales will be important.The exact equilibrium distribution for the axion is, up to normalization, ρ ( θ, H I ) ∝ exp (cid:18) − π χ ( H I / π )(1 − cos θ )3 H I (cid:19) (16)for H I (cid:46) πT c , about 800 MeV, the susceptibility is constant: ρ ( θ, H I ) ∝ exp (cid:18) − π χ (0)(1 − cos θ )3 H I (cid:19) (17) ≈ exp (cid:18) − − cos θ ( H I /
170 MeV) (cid:19) (18)This distribution is known as the von Mises distribution. It is plotted in Fig. 1 for H I ranging from 100 to 240 MeV. H I
100 MeV H I
135 MeV H I
170 MeV H I
205 MeV H I
240 MeV - - - θ ρ ( θ , H I ) H I
100 MeV H I
135 MeV H I
170 MeV H I
205 MeV H I
240 MeV θ ρ ( log θ , H I ) FIG. 1: Equilibrium distribution for the QCD axion, in terms of θ (left) and log | θ | (right), at various H I For H I (cid:28) (8 π χ (0) / / , about 170 MeV, this distribution is approximately a narrow Gaussian with width( H I /
170 MeV) : ρ ( θ, H I ) ∝ exp (cid:18) − θ H I /
170 MeV) (cid:19) (19)The distribution ρ (log | θ | ) is also plotted. This distribution is peaked where θ is comparable in magnitude to thewidth of the original distribution. Note that, in the Gaussian (low H I ) regime, this distribution falls off slowly (powerlaw in θ ) at small log | θ | , and quickly (exponentially in θ ) at high log | θ | . The left tail corresponds to the center of theGaussian, where the distribution is smooth, with the power law coming from the change of measure on θ ; the righttail inherits its exponential suppression from the tails of the Gaussian.This approximation breaks down as the width become O (1) and “sees” the anharmonic shape of the potential. For H I (cid:29)
170 MeV, the probability distribution becomes flat, with a small sinusoidal variation: ρ ( θ, H I ) ∝ − − cos θ ( H I /
170 MeV) (20)The relative variation in ρ , across its range, is less than .001 for H I > πT c , so we do not need to explicitly modelthe temperature-dependence of χ in this part of our analysis.In general, we will refer to H I (cid:46)
170 MeV as “low H I ” and H I (cid:38)
170 MeV as “high H I ” for the remainder of thepaper.As discussed in Sec. VI A, the axion density is determined by the value of H at (cid:15) ∼ m /H = ⇒ ˙ H ∼ m . Thiscan be higher than the value of H in the last 60 e-folds (the value observable in our universe).Because the axion density is a function of both the PQ scale f a and the initial (post-inflation) value of θ , requiringaxions to make up all of dark matter (Ω a = Ω c ) imposes a relationship between f a and θ . We will use θ DM ( f a ) torefer to the initial θ needed for a given f a , subject to this constraint. For more details on the functional form of θ DM ( f a ), see Appendix A.
1. The Stochastic Axion Window
For any value of H I and f a , the value of θ in a given patch will be random, following the distribution ρ . In orderto make our model natural, θ DM ( f a ) should not be an unusually high or unusually low value for this distribution.We can quantify this in terms of the probability p that | θ | ≤ θ DM , and the corresponding probability q = 1 − p that | θ | > θ DM . There are two ways for the observed θ to be “unnatural”: it can be unnaturally large (very small p )or unnaturally small (very small q ). Note that this is a very concrete notion of naturalness: we have a finite set ofHubble patches after inflation, with a known distribution of misalignment angles. In particular, p is a fraction ofHubble patches, rather than an abstract or epistemic notion of probability.Technically, p ( θ ) is the CDF of the distribution ρ ( | θ | ). In order to plot contours of f a and H I for fixed p , we needthe inverse CDF θ ( p ). This is slow to compute numerically, particularly for extremely small values of q , but theGaussian distribution for a noninteracting particle of the same mass is a good approximation in both limits. (Forlarge H I , a very wide Gaussian truncated at | θ | = π is approximately uniform, as desired). For values of H I around144 MeV we can interpolate numerically as long as q is not too small.The axion parameter space is shown in Fig. 2. For high f a , the value of Hubble during inflation depends on f a and is confined to about an order of magnitude (in order for p to fall between 0.1 and 0.9). For higher H I , it isstill possible to obtain the correct dark matter density via anthropics, with fine-tuning ( p ) of no more than 10 − ; forlower values of H I , however, the value of q drops off exponentially due to the tails of the Gaussian, and implausibleamounts of fine-tuning are required to produce enough dark matter. Therefore, long periods of extremely-low-scaleinflation are incompatible with the anthropic axion. The solid contours are for p or q equal to 10 − , − , − , − .The dashed contours are for q = 10 − , − , , − , , and use the Gaussian approximation without numericalinterpolation.In short, we have carved up the former “anthropic axion” window into three regimes: for high f and high H I wehave a concrete model of an anthropic axion with a uniform prior probability, for very low H I we have a low axionabundance with θ extremely small, and in the middle there is a band where axion dark matter is naturally obtained,which we call the stochastic window. In this middle region (the stochastic window) f a ranges from ≈ GeV tothe Planck scale. Within this window, two observables become correlated: the Hubble scale of inflation and the massof a dark-matter QCD axion. Lighter axions (higher f a ) require a lower Hubble scale.The red region is ruled out due to a backreaction effect, which concentrates ρ around the maxima of V , overproducingdark matter. This comes into play when the relaxation time for the axion is too long. This rules out a decay constant f a > (cid:112) / M P ≈ . × GeV, or a mass m a (cid:46) . × − eV, see Sec. VI D and Appendix B. The blue region isconsistent with our model, but the relaxation time is longer than inflation can last without becoming eternal; in thiswindow, either inflation is eternal or it is so short that the misalignment angle is set by initial conditions. Also shownare observational constraints due to isocurvature [48, 49] and supernova 1987A [51–53]. The gray region shown athigh f a is disfavored due to black-hole superradiance [54]. There is an upper bound on H I (during the last 60 e-folds)from Planck and BICEP2/Keck constraints on primordial B -modes [55, 56].For other analyses of ( H I , f a ) parameter space, see e.g. [48, 49, 57–66]. Our results agree with these, up to the factthat we have added the allowed dark matter region to the left of the isocurvature bound of course. p = .1 p = .01 p = .001BH Superradiance Bound P l an ck ( ) B ound q = .1 q = .0001 q = - q = - q = - Isocurvature BoundSupernova 1987A BoundEquilibrium only if Eternal InflationBackreaction ClassicalWindowStochastic Window10 - - - - - GeV10 GeV10 GeV10 GeV10 GeV10 GeV10 GeV10 GeV10 GeV10 GeV10 GeV10 GeV 10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV10 - eV kHzMHzGHzTHz H I ( GeV ) f a E I ( GeV ) m a Inflationary Axion Parameter Space
FIG. 2: Parameter space for the QCD axion dark matter, assuming a long enough period of inflation that the axion reachesequilibrium as described in the text. Axes are axion decay constant f a (left) and mass m a (right, inverted), Hubble scale ofinflation H I (bottom), and inflationary energy scale E I = (3 H I M P ) / (top).In the large green region, the observed dark matter density is a typical density to get from our axion equilibrium distribution( p > . q > . p and q are shown as solid and dashed contours around this region. At near-Planckian f a ,the axion’s behavior changes: in the pink region, backreaction effects become significant and force θ → π ; in the blue region,the distribution does not reach equilibrium and depends on initial conditions, except in eternal inflation.At high H I and low f a is the classical window, where PQ symmetry breaks and produces axions after inflation. The thin greenline shows the standard value of f a where this production matches the observed dark matter density.Observational constraints are shown in gray: isocurvature from the CMB spectrum, a lower bound on f a from supernova 1987A,black-hole superradiance, and an upper bound on H I from the Planck 2015 constraint on r .
2. Classical Window
The fate of the axion depends on whether the universe reheats to a temperature above f a . The reheating temperaturedepends on the efficiency of reheating, (cid:15) eff ≤ T rh = (cid:15) eff E I (21)where E I = (3 H I M P ) / is the energy scale of inflation.If T rh > f a , PQ symmetry is restored, the misalignment angle is destroyed, and axions are produced througha different mechanism when the universe cools and PQ breaking occurs. Even if reheating does not reach thistemperature, the de Sitter temperature during inflation could, if H I / π > f a . In this case, PQ symmetry is maintainedthroughout inflation, so no axions and no isocurvature fluctuations are produced to begin with. These two lines formthe right edge of the “Isocurvature Bound” region in Fig. 2, with (cid:15) eff = 10 − . for the sake of illustration. Thesame parameter space is shown in Fig. 3 with (cid:15) eff (cid:28) (cid:15) eff = 1 (maximally efficientreheating).In the region to the right of this boundary, PQ symmetry is preserved or restored until after reheating. Once theuniverse cools below f a , the process of symmetry breaking randomizes the axion and gives it an effective misalignmentangle θ C everywhere. This implies a definite value f a = f C for which there is the correct abundance of dark matter( θ C = θ DM ( f C )), which we refer to as the classical axion window and show as the thin green line on the right sidein Fig. 2. There is significant systematic uncertainty about this value, due in part to the difficulty of accounting foraxion string decay contributions, which increase θ C . BH Superradiance Bound P l an ck ( ) B ound Isocurvature BoundSupernova 1987A BoundEquilibrium only if Eternal InflationBackreaction ClassicalWindowStochastic WindowOverproductionUnderproduction10 - H I ( GeV ) f a ( G e V ) E I ( GeV ) Inflationary Axion Parameter Space
BH Superradiance Bound P l an ck ( ) B ound IsocurvatureBoundIsocurvatureBound Supernova 1987A BoundEquilibrium only if Eternal InflationBackreaction Classical WindowStochastic WindowOverproductionUnderproduction10 - - - - - - - - - - - - - H I ( GeV ) E I ( GeV ) m a ( e V ) Inflationary Axion Parameter Space
FIG. 3: The same parameter space as Fig. 2 except with maximally inefficient (left) or maximally efficient (right) reheating.The main difference is that the isocurvature bound moves and so does the boundary between the classical and stochasticwindows. If reheating is inefficient, then the axion is produced after inflation if T dS = H I / π > f a . If it is maximally efficient,then the weaker bound E I > f a holds instead. In general, the bound is given by max( T dS , T rh ) > f a where T rh = (cid:15) eff E I isthe reheating temperature. Suppose f a is slightly higher than f C . Naively, the correct dark matter abundance could still be obtained after PQsymmetry breaking if the axion accidentally ended up near the (eventual) minimum of its potential by random chance.However, this feat would need to be replicated independently in many causally-disconnected Hubble patches; a singleHubble patch today contains over 10 regions that were Hubble patches at PQ breaking. Therefore, the probabilityof an effective misalignment angle less than θ C /
10, in each of these patches, is 10 − . This is an absurdly smallnumber even compared to the “dead” axion at very low H I . The same conclusion applies in the other direction, with f a < f C ; the classical axion window is extremely narrow. B. Fuzzy Dark Matter
This same stochastic mechanism for producing dark matter abundance from quantum fluctuations of a field duringinflation can apply more broadly than just to the QCD axion. In this section we consider a general scalar field, andin particular motivated by fuzzy dark matter (see e.g. [67]).
1. Free Scalar Model
For the case of a free scalar field, our mechanism works to produce the observed dark matter density over a widerange of masses. The required Hubble scale is roughly H I ≈
500 eV (cid:16) m − eV (cid:17) / (22)= 500 GeV (cid:16) m
100 eV (cid:17) / (23)There is a lower bound on mass from the backreaction and eternal-inflation bounds (see Sec. VI), which are both parametrically H I (cid:46) mM P in this case. This rules out m (cid:46) − eV, but also cuts into the stochastic window forany m (cid:46) − eV. In this case, the backreaction behaves differently than for axions; rather than sitting on a hilltop,the field runs away to extremely large field values, stopping only when it backreacts significantly on spacetime andbegins to act as an inflaton.The upper bound is set by isocurvature (without anharmonic corrections), which rules out masses greater thanaround 100 MeV. At this upper bound, H I ≈
50 TeV.This leaves essentially the entire range where weakly-coupled scalar dark matter is of interest. In particular, fuzzydark matter ( m ≈ − eV) is allowed. The observed abundance of dark matter can be reached naturally for a0quadratic fuzzy dark matter (FDM) potential with H I of 200 −
800 eV. This is the stochastic window for FDM (but H I >
600 eV requires eternal inflation to reach equilibrium).
2. Axionic Model
For fuzzy dark matter, one common assumption is a non-QCD axion with decay constant f a ≈ GeV and auniformly random initial θ [67].For an axion-like potential, we can easily accommodate higher f a by adjusting H I downwards slightly to producea small θ ; on the other hand, for f a ≈ GeV, any H I (cid:38) θ . The usualbackreaction bound for axions, f a (cid:46) M P , applies in this case; the backreaction bound for a free scalar assumes thatthe maximum height of the potential is comparable to the inflaton. Therefore, large values of H I are permitted, upto the isocurvature limit.For even lighter axion-like particles, the spatial variation of the ALP field (caused by inflationary fluctuations inthe last 60 e-folds) can rotate CMB E modes into B modes. This places a bound on the ALP-photon coupling atthese very low masses [68, 69]. VI. INFLATIONARY SECTOR
In the analysis above, we have made several simplifying assumptions about the background spacetime. In essence,we have been working with de Sitter – an eternally-expanding non-dynamical background geometry with constant H –rather than an inflationary FLRW universe, where H ( t ) changes, inflation may last for only a finite duration, and thespectator field may backreact on the spacetime geometry. Taking this all into account produces several modificationsof the picture above.First, the H I we discuss above is not necessarily the value of H observed from the last 60 e-folds, and can besignificantly higher. A freeze-out-like mechanism, based on the slow-roll parameter rather than temperature, sets theapplicable H I .Second, inflation may not last long enough for the dark matter field to reach its equilibrium distribution. Wecalculate the number of e-folds required for the QCD axion, and find a regime where the relaxation time is longerthan can be achieved without eternal inflation. If inflation does not last long enough for significant relaxation, thedark matter density is set by initial conditions.Finally, the dark matter field may backreact. In particular, patches with higher V ( φ ) will expand at a slightlyfaster H , which allows them to outnumber the lower- H patches if the relaxation time is long enough. We find thatthis occurs for low H I and nearly-Planckian f a for the QCD axion; in this regime, the backreaction effect pushes theaxion towards θ = π , where the faster exponential growth of space outpaces the relaxation process. A. Variation in H I In this section, we drop the assumption that H I is constant over time.The equilibrium distribution for a massive scalar gives an expectation value for the field’s energy density, (cid:104) V ( φ ) (cid:105) ,of order H I . The relaxation time is of order H I /m , or H I /m e-folds. In an inflating universe, as opposed to deSitter, H is a function of time. In order for the field to approach equilibrium faster than the equilibrium density ischanging, we need d log Hdt (cid:46) m H (24)˙ H (cid:46) m (25) (cid:15) (cid:46) m H (26)where (cid:15) = ˙ H/H is the first slow-roll parameter. Therefore, the field stays in equilibrium as long as the inflaton isrolling slowly enough, (cid:15) (cid:28) m H , and “freezes out” at (cid:15) ≈ m H . The energy density of the field after reheating will be oforder H ∗ , where H ∗ is the Hubble scale at the last time that ˙ H (cid:46) m . (This may occur multiple times, because ˙ H doesn’t have to change monotonically.) Any field with m below the minimum value of ˙ H will be “frozen” the entire1time and remain at its initial field value. If inflation is never chaotic, then ˙ H > H /M P throughout, so any field with m < H /M P will fall into this category.Consider the case of multiple scalar fields. Because H decreases monotonically, the lightest weakly-coupled scalarwill have the highest density after reheating. In many low-scale inflation models (such as hilltop models) H is nearlyconstant, so the densities at reheating will be of the same order of magnitude (the exact densities are random). Lightfields begin to dilute later, giving an extra factor of m − / to the density. (This assumes that all of the scalars havemasses above H eq ≈ − eV, the Hubble at matter-radiation equality, and that their potentials do not change afteroscillation begins). Therefore, the lightest scalar will almost always be the dominant component; if the second-lightestis only a few times heavier, however, it is not too unlikely for it to have a higher density through random chance.If there are a sufficient number of scalars over a range of masses whose abundances can be measured, then this m − / dependence could provide an observable signature that distinguishes the stochastic scenario from anthropic selection,which generically causes all scalars to have comparable densities (except for scalars that do not have overclosureproblems in the first place, e.g. axions with low f a ). B. Length of Inflation
In this section, we discuss how long inflation needs to last for ρ to achieve equilibrium.For H I (cid:46)
170 MeV, as shown in Sec. V A, we can treat the axion as a free scalar. In this case, the equilibriumdistribution is reached on a timescale of 3 H I /m . Therefore, for low H I , the production mechanism and parameter-space analysis above are relevant when the number of e-folds is N (cid:29) H I /m , where N r = 3 H I /m is the relaxationtime in e-folds. For a QCD axion, with H I in the MeV − GeV range, this is anywhere from about 10 to 10 e-folds. This corresponds to a relaxation time ranging from one week (at f a = 10 GeV) to several million years(at f a ≈ M P ). While much higher than the 50-60 e-folds required by the horizon problem, this is achievable given aflat enough potential. Note that models such as the relaxion to solve the hierarchy problem require a similarly lowHubble scale and even longer periods of inflation (see e.g. [36]), which can provide some motivation for consideringsuch inflationary sectors.For H I (cid:38)
170 MeV, we should treat the potential as flat, and need to take the axion’s compact range into account.As noted in Sec. III C and Appendix B, the relaxation time in e-folds is N r = 8 π f a /H I In this regime, the stochasticwindow is restricted to f a (cid:46) GeV, so N r is at most 10 or so.These relaxation times place a lower bound on the number of e-folds N (cid:38) N r necessary for our model to beindependent of initial conditions. This can be converted to a slow-roll parameter (cid:15) . This parameter can be expressedas − d log H/dN , so the dark matter field is in equilibrium as long as, at some point during inflation, (cid:15) remains (cid:46) /N r over at least an order of magnitude in H .There is also an upper bound on N for any non-eternal inflationary model, and an upper bound on N r which avoidsa backreaction effect. These are discussed in the following two sections. C. Eternal Inflation
One way to obtain very long periods of inflation is via eternal inflation. This gives rise to an infinite volume of theuniverse, with accompanying measure problems. If we avoid eternal inflation, then an initial Hubble patch evolves intoa finite population of Hubble patches at reheating, and the distribution ρ ( θ ) of misalignment angles can be interpretedmore easily.There is actually a sharp upper bound on the number of e-folds of inflation that can be obtained while still remainingin the non-eternal regime, given by[70] N < π M P H I (27)We derive this from a related bound in Appendix C.Given this constraint, in the free scalar case, it is possible to maintain classical rolling for enough e-folds to reachequilibrium as long as 3 H I /m < π M P / H I , which is equivalent to m > H I √ πM P (28)2For the QCD axion at low H , this implies f a M P < √
23 Λ QCD H (29)where Λ QCD = χ (0) / . This constraint is satisfied in the H I (cid:46)
170 MeV branch of the stochastic window.For higher H I , our constraint is instead 8 π f a /H I < π M P / H I , which is equivalent to f a M P < √ H I (cid:38)
170 MeV branch of the stochastic window. Note that this is of order M P like thebackreaction bound, but for high H I instead of low H I ; together, they rule out transPlanckian or nearly-Planckian f a in the case of non-eternal inflation.In the region above the stochastic window, these two constraints are not always obeyed. We can compute amore precise constraint, valid for all H I including H I ≈
170 MeV, by finding the fastest nonzero decay rate of aquasinormal mode of the Fokker-Planck equation (see Appendix B); we do this numerically, and confirm that itmatches our analytical results in both regimes. The result is the region labeled “Eternal inflation” in Fig. 2; withinthis region, inflation must either violate the classical-rolling constraint (leading to a finite probability of an infinitereheating volume, i.e., eternal inflation) or have a duration of less than one relaxation time, in which case our resultsdo not apply and the misalignment angle is simply determined by initial conditions prior to inflation. In either case,our concrete notion of fine-tuning – based on a finite set of Hubble patches for which the distribution of field valuesis calculable – is inapplicable.
D. Backreaction
In this section, we drop the assumption that the field does not backreact significantly on spacetime and find inwhat region backreaction is significant and cannot be neglected.The height of the axion potential, χ ≈ Λ QCD , is much smaller than the inflaton’s energy density 3 H I M P for ourentire parameter space. However, this does not mean that backreaction effects can be ignored. Although the changein Hubble scale ∆ H between θ = 0 and θ = π is tiny, it can add up significantly over a long relaxation time t r : thenumber of patches with θ ≈ π will be enhanced by up to exp(3∆ Ht r ). Over several relaxation times, each patchexperiences the same distribution of values for θ and H , so this effect does not compound for much longer than t r .We can get a quick parametric estimate of when 3∆ Ht r (cid:38) H ≈ V max ( φ ) / (2 M P H I ) ≈ χ/ (2 M P H I ). Therefore, our bound is roughly N r (cid:46) M P H I χ (31)The total number of e-folds of inflation can be larger than this; it is a bound only on the relaxation time.For low H I , we have t r ≈ H I /m a , so we should be concerned when 3∆ Ht r ≈ χ/ (2 M P m a ) (cid:38)
1. We have χ ≈ m a f a for low H I , so backreaction effects impose a bound on f a which is roughly f a (cid:46) √ √ M P (32)For H I (cid:38)
170 MeV, we have t r ≈ π f a /H I , so that 3∆ Ht r ≈ χ π f a / ( M P H I ). For H I (cid:46)
800 MeV, χ ≈ χ (0) ≈ m a f a , so backreaction effects kick in around ( χ (0)4 π f a ) / ( M P H I ) ≈
1, which imposes the bound (cid:18) f a M P (cid:19) (cid:46) (cid:18) H I
170 MeV (cid:19) (33)As H I rises above 170 MeV, this bound rapidly passes the Planck scale. By the time we reach H I ≈
800 MeV, where χ begins to decrease, the backreaction effects are negligible for any f a (cid:46) M P ; the decrease in χ makes them weakerstill, so we can again ignore the temperature-dependent axion mass.For a more detailed analysis of backreaction effects, which confirms these rough estimates and gives them moreprecise meaning, see Appendix B. In Fig. 2, the region ruled out by backreaction is plotted by setting N = 1 /t r and calculating t r numerically via the eigenvalue method described in Appendix B. This roughly agrees with the twoapproximate limits in their respective regions from equations (32) and (33). In this region, for a long enough periodof inflation, axion dark matter is significantly overproduced.3 VII. CONCLUSIONS
We have found that, if inflation happens at a low scale, the minimal QCD axion is naturally driven towards anequilibrium distribution during inflation. This equilibrium is between the classical force driving the axion to theminimum of its potential and the stochastic quantum fluctuations of the axion field. With a sufficiently long periodof such low-scale inflation, the axion will reach its equilibrium. This equilibrium is independent of the precise lengthof inflation and depends only on the axion mass m a and the Hubble scale of inflation H I . Then we can predictthe approximate size (officially the probability distribution) of the ‘initial’ axion misalignment angle θ after inflationand thus the final axion abundance today. This equilibrium prediction is shown in Fig. 2. The region in which theaxion achieves roughly the measured dark matter abundance is shown as the broad green band. In the classical axionwindow (the thin green line on the right of the figure), the axion abundance is determined precisely by the axionmass. In our new region the axion mass and H I determine only the rough size of the axion abundance but it can stillvary by O (1) (part of why the region is broad). Thus we can see that the minimal QCD axion model can naturallyproduce the correct dark matter abundance for a wide range of axion masses from roughly 10 − eV, a little abovethe classical axion window, all the way down to the lowest mass around 10 − eV. This corresponds to axion decayconstants from f a ∼ GeV all the way up to almost the Planck scale.This axion equilibrium will arise from a sufficiently long period of normal inflation, or could also arise if there was aperiod of eternal inflation in our past (e.g. [71]). Eternal inflation (followed perhaps by some period of normal inflation)would certainly be long enough to produce an equilibrium distribution for the axion, however it also famously comeswith measure problems and so it might not be possible to use our predicted probability distribution for the axion.Interestingly, in our mechanism the axion abundance arises from the stochastic quantum fluctuations of the axionfield during inflation, but nevertheless there are no observable isocurvature fluctuations induced. The reason is thatover a long period of inflation the axion field spreads out significantly in its potential well, but each individual quantum‘jump’ of the field is quite small. Thus, in the last 60 or less e-folds (those relevant for the observable universe today)the axion spreads out only a very little distance in field space in its potential and so the isocurvature perturbationsof wavelengths smaller than today’s horizon size are quite small. This same mechanism can also produce a nonzeroabundance of other scalar fields (e.g. for fuzzy dark matter) from quantum fluctuations during inflation withoutproducing dangerous isocurvature perturbations.Thus we see that just the minimal QCD axion model can naturally reproduce the observed dark matter abundancedown even to the lowest masses ( f a almost up to the Planck scale). This motivates searching experimentally for QCDaxions broadly over the entire mass range. Acknowledgements
We acknowledge useful conversations with Savas Dimopoulos, Michael Dine, David E. Kaplan, Andrei Linde, SurjeetRajendran, Leonardo Senatore, and Eva Silverstein. This work was supported by NSF grant PHY-1720397, DOEEarly Career Award de-sc0012012, Heising-Simons Foundation grant 2015-037, and the Mellam Graduate Fellowship.Note added: Shortly after this paper appeared, another overlapping paper appeared which generally agrees with ourresults [72].
Appendix A: Misalignment Angle and PQ Scale f a Depending on f a , there are three regimes for the behavior of the misalignment mechanism for a QCD axion. Inall cases, the axion is frozen by Hubble friction until 3 H (cid:46) m a , after which it oscillates and behaves as cold darkmatter. The misalignment angle needed for a correct abundance of axion dark matter, θ DM ( f a ), can be calculated byextrapolating the axion density backwards from the present to the time of oscillation. θ DM is a roughly power-lawfunction of f a , ranging from about 10 − at f a ∼ M P to O (1) for f a ∼ GeV.At high f a (low m a ), the axion is frozen until after the QCD phase transition. In this case, the density now is justthe density at oscillation, diluted by a factor of a :Ω a ∼ m a φ a (A1)In a radiation-dominated universe, a ∝ H − / , so the density is proportional to Λ QCD θ H − / . Noting that H osc ∼ m a ∼ f − a , the density goes as θ f / a . Holding this constant, we have that θ DM ∝ f − / a .At lower f a (cid:46) GeV, when the axion oscillates before the QCD transition, the mass at the time of oscillation m a, osc is a power-law function of the temperature T osc of the quark-gluon plasma. This introduces an additional factor4of f na into θ DM , where the exponent n can be derived analytically (e.g. dilute instanton gas approximation (DIGA))or numerically via lattice methods (e.g. [50, 73, 74]). For example, DIGA (with three light quarks, to leading order)gives m ∝ T − and θ DM ∝ f − / a [73]. However, we do not use DIGA. Instead, we combine results from the moreaccurate numerical studies, which produce various exponents n in the vicinity of − /
12. There is a wiggly bend in θ DM at f a ≈ GeV, corresponding to a dark-matter axion which begins to oscillate at the QCD transition.At even lower f a ≈ GeV, the axion must begin with a misalignment angle close to π . At this point, anharmoniccorrections must be included; one paper[48] gives the functional form in this regime as θ DM = π − exp( Af a + B ) forsome constants A, B .To get a more precise model for θ DM , we fit a nonlinear function to the f a vs. θ DM curves from several paperswhich use various techniques to deal with these three regimes[48, 50, 57]. There is a disagreement in the literatureregarding the overall prefactor for f a in this relation, so we multiplied the values of f a from [57] by 1 .
133 and from[48] by 1 .
676 to agree with [50] (the most recent work) in the region of overlap of the curves. (These values wereobtained from a numerical fit.)First, we transformed θ nonlinearly to a new variable X = (cid:18) θπ (cid:19) (cid:18) − C θπ − log (cid:18) π − θπ (cid:19)(cid:19) (A2)where the constant C ≈ .
267 was fit by hand. X is linear in θ at θ (cid:28)
1, and logarithmic in π − θ at θ ≈ π . We fit asmoothed piecewise power law for f a ( X DM ), with a power of − / f a , − f a , and an arbitrary power(the fit produced − . − / f a ( θ DM ) which is power-law at small angles, with a change inthe exponent around 10 GeV (which fits better than 10 due to the “wiggliness” of the transition) and linear inlog( π − θ ) at low f a , which accounts for anharmonic effects when the axion is on the “hilltop” at θ ≈ π [48].The fitted function agrees well with the (shifted) curves from [48, 50, 57]: within 5% except where the source papersdisagree by about 20% near the 10 GeV transition.
Appendix B: Fokker-Planck Formalism and Inflationary Backreaction
This appendix follows the approach of [75], specialized to the QCD axion potential and with the additional intro-duction of the backreaction effect.Consider a slow-rolling scalar field φ in an expanding spacetime. As in the body of the paper, we consider thelong-wavelength modes separately from the short-wavelength modes and focus on the former. Starobinsky[38] showedthat these evolve classically, according to the usual slow-roll equation with an additional stochastic (random walk)term: ˙ φ = − V (cid:48) ( φ )3 H I + f ( t ) (B1)where f a is Gaussian noise with correlation function (cid:104) f ( t ) f ( t ) (cid:105) = H I π δ ( t − t ) . (B2)This is a Langevin equation, which describes the evolution of φ over time as a stochastic random variable. It is moreconvenient to work in terms of the probability density ρ ( φ, t ), which gives us a (deterministic) Fokker-Planck equation:˙ ρ ( φ, t ) = 13 H I ∂ φ ( V (cid:48) ( φ ) ρ ( φ, t )) + H I π ∂ φφ ρ ( φ, t ) (B3)Any initial probability distribution will asymptotically approach an equilibrium distribution of the form: ρ f ( φ, t ) ∝ exp (cid:18) − π V ( φ )3 H I (cid:19) (B4)In the body of the paper, we derive parametric estimates of the relaxation time for the axion distribution ρ and ofthe scale f a where backreaction effects become significant, in the small and large H I regimes. We can obtain moreprecise descriptions of these quantities by decomposing the time evolution of ρ into quasinormal modes. The shapesand half-lives of these modes are the eigenfunctions and eigenvalues of a Schr¨odinger-like equation with a potentialthat is related to the axion potential.5To understand the backreaction effect, we will momentarily work in terms of P ( φ, t ) = e Ht ρ ( φ, t ) (B5)This distribution counts the total number of patches with average field value φ , instead of the relative frequency.Whereas the integral (cid:82) ρ d φ is always 1, the integral of P grows as the universe expands. We can write a Fokker-Planck equation for P by including a term for this growth:˙ P ( φ, t ) = 13 H ∂ φ ( V (cid:48) ( φ ) P ( φ, t )) + H π ∂ φφ P ( φ, t ) + 3 HP ( φ, t ) (B6)However, H is not truly independent of φ . Patches where the field is farther up its potential will have a slightly higherenergy density V tot = V I + V ( φ ), where V I is the energy density due to the inflaton and V is the potential for φ . (Ofcourse, this decomposition depends on our choice of zero for V ( φ ); we choose V (0) = 0, so that V ( φ ) ≥ H = (cid:112) V tot / M P will be slightly larger as well. As long as V (cid:28) V I , we can linearize: H ( φ ) = H I + ∆ H ( φ ) (B7) ≈ H I + V ( φ ) / (6 M P H I ) (B8)where H I = (cid:112) V I / M P is the contribution from the inflaton alone. Decomposing the last term of our Fokker-Planckequation in this way,˙ P ( φ, t ) = 13 H ∂ φ ( V (cid:48) ( φ ) P ( φ, t )) + H π ∂ φφ P ( φ, t ) + 3 H I P ( φ, t ) + 12 M P H I V ( φ ) P ( φ, t ) (B9)we see that there is a new φ -dependent term due to this backreaction. At this point, we would like to return to ρ ,but we encounter a subtle difficulty. We want ρ to be normalized to 1, while remaining proportional to P at a giventime. The appropriate definition is P ( φ, t ) = e H avg t ρ ( φ, t ) (B10)where H avg is the average growth rate of all patches, given by H avg ( t ) = (cid:90) H ( φ ) ρ ( φ, t )d φ (B11)While P is described by a local differential equation, this integral means that ρ is not! The dynamics of ρ in asmall interval of φ depend on the average growth rate, which in turn depends on what ρ looks like globally. We willcompromise slightly and define an unnormalized distribution (cid:101) ρ by P ( φ, t ) = e H I t (cid:101) ρ ( φ, t ) (B12)The full time-evolution of (cid:101) ρ is given by the Fokker-Planck equation above, with the 3 H I term removed:˙ (cid:101) ρ ( φ, t ) = 13 H I ∂ φ ( V (cid:48) ( φ ) (cid:101) ρ ( φ, t )) + H I π ∂ φφ (cid:101) ρ ( φ, t ) + 12 M P H I V ( φ ) (cid:101) ρ ( φ, t ) (B13)The last term in this equation is due to the backreaction effect.Note that (cid:101) ρ is not a probability distribution: the integral (cid:82) (cid:101) ρ d θ will change over time at a rate H avg − H I , which isjust the average of ∆ H . This normalizes away the growth due to the inflaton but leaves the extra contribution frombackreaction. (cid:101) ρ is useful because it reduces to ρ when backreaction is negligible, while still having local dynamics in φ . We now substitute (cid:101) ρ ( φ, t ) = Ψ( φ ) ψ ( φ, t ), whereΨ( φ ) := exp( − ν ( φ )) (B14) ν ( φ ) := 4 π H I V ( φ ) (B15)in order to rewrite this diffusion equation as follows, in terms of ψ : − π H I ˙ ψ ( φ, t ) = − ψ (cid:48)(cid:48) ( φ, t ) + 12 (cid:20) − ν (cid:48)(cid:48) ( φ ) + ν (cid:48) ( φ ) − M P ν ( φ ) (cid:21) ψ ( φ, t ) (B16)6This is a Wick-rotated time-dependent Schr¨odinger equation. It has eigenfunction solutions which decay exponentially, ψ ( φ, t ) = (cid:88) i c i e − Γ i t ψ i ( φ ) (B17)The eigenvalues Γ i and eigenfunctions ψ i are given by the corresponding time-independent equation4 π H I Γ i ψ i ( φ ) = − ψ (cid:48)(cid:48) i ( φ ) + 12 (cid:20) − ν (cid:48)(cid:48) ( φ ) + ν (cid:48) ( φ ) − M P ν ( φ ) (cid:21) ψ i ( φ ) (B18)So far, everything we have said applies to a generic scalar field. For a quadratic potential V = m φ , this equationbecomes ν ( φ ) = 2 π m H I φ (B19)4 π H I Γ i ψ i ( φ ) = − ψ (cid:48)(cid:48) i ( φ ) + π m H I (cid:20) − (cid:18) π m H I − M P (cid:19) φ (cid:21) ψ i ( φ ) (B20)For H I (cid:28) M P m , the backreaction term 3 /M P is negligible. However, if H I > π M P m , the sign of the “potential”flips over, creating an instability. Physically, this means that the patches where the field is further up its potentialexpand faster by a large enough ∆ H to outpace the relaxation process, so the distribution of patches runs away toextreme field values. If nothing intervenes, this will continue until ∆ H ≈ H I , at which point the field φ becomes asecond inflaton field.For an axion potential, we substitute φ = f a θ and find ν ( θ ) = 4 π χ H I (1 − cos θ ) (B21)4 π f a H I Γ i ψ i ( θ ) = − ψ (cid:48)(cid:48) i ( θ ) + 12 (cid:2) − α (1 − β ) cos θ + α sin θ − αβ (cid:3) ψ i ( θ ) (B22)where α := 4 π χ H I = 12 (cid:18) H I
170 MeV (cid:19) − (B23) β := 3 f a M P (B24)The eigenfunctions ψ i correspond to quasinormal modes given by (cid:101) ρ i = Ψ ψ i . The eigenvalues Γ i of these functionsare not energies, but decay rates. With the backreaction term neglected, we always have Γ = 0 and ψ = Ψ; thedistribution (cid:101) ρ = Ψ ψ = Ψ is the equilibrium state of the diffusion equation. With backreaction, (cid:101) ρ still gives thebehavior at late time, but Γ < V ( φ ). We chose V ( φ ) ≥
0, so the backreaction is always a positive contribution to growth and Γ is negative.For a different choice, our Γ i would all shift by some constant.) The gap Γ − Γ gives the rate for (cid:101) ρ to decay relativeto (cid:101) ρ . This is the slowest-decaying mode, so 1 / (Γ − Γ ) is the relaxation time.Note that the “potential” here is not the axion potential. It has a term proportional to − V (cid:48)(cid:48) ∝ V ∝ cos θ , but alsoanother ∝ sin θ with twice the frequency. In addition to the minimum at θ = 0, this produces another minimum at θ = π .Setting aside the backreaction effects momentarily, we can use this formalism to compute more precise relaxationtimes. For low H I ( α (cid:29) term dominates and we can approximate the system as a simple harmonicoscillator. This approximation gives Γ = m a / H I and Γ n = ( n + ) m a / H I , with a relaxation time of 3 H I /m a asexpected. (Of course, we should really have Γ = 0; it is easy to check that the true ground state ψ ∝ exp( − ν )has Γ = 0. The harmonic-oscillator approximation gives the correct level spacing but not the correct ground-stateenergy.)For high H I , the potential is negligible; the eigenfunctions are ψ n ( θ ) ∝ cos( nθ ). It is easy to see that ψ is constant,Γ = 0, and Γ = H I / π f a . This gives a relaxation time of 8 π f a /H I , which agrees with the estimate in the bodyof the paper up to a factor of 2.For β (cid:28) f a (cid:28) M P ), the effect of backreaction is negligible and the θ = 0 minimum dominates. In this regime,the ground state ψ of the double-well potential is still approximately Ψ. As β approaches 1, the two minima become7closer. At β = 1, the cosine term in the Schr¨odinger potential vanishes and they are exactly degenerate. For β > θ = π minimum is the global minimum.This crossover has several effects. First, the ground state becomes a mix of the two minima, then shifts to the π min-imum. At β = 1 ( f a = M P / √ ψ (0) = ψ ( π ). However, this is not the phenomenologically-relevant transition: the factor of Ψ still ensures that (cid:101) ρ (0) (cid:29) (cid:101) ρ ( π ). The important transition is at β = 2, where thecosine’s coefficient (1 − β ) is the negative of its value in the f a (cid:28) M P limit. It is easy to see that this is equivalent tosending ν (cid:55)→ − ν or θ (cid:55)→ π − θ , so that ψ ∝ exp( ν ( θ )) ∝ / Ψ, and (cid:101) ρ = Ψ ψ =constant. Therefore, the equilibriumdistribution at β = 2 ( f a = M P √ / √
3) is uniform. (A numerical investigation of the ground state for general β sug-gests that, for β approaching this value, the ground state is approximately a mixture of Ψ and a uniform distribution,with the proportions of the mixture changing smoothly.) Above β = 2, for low H I ( α (cid:38) θ = π once the peak in ψ becomes more significant than the valley in Ψ.Second, the splitting between the lowest two states becomes small (but nonzero), so the relaxation time becomeslong. (Phenomenologically, this means that for a narrow window around f a = M P / √ f a = M P √ / √
3, the relaxation time returns to thevalue it would have without the effects of backreaction; for higher f a it continues to drop.For low H I (high α ), this crossover happens extremely quickly, with the splitting becoming extremely small. As H I → f a = M P / √
3. (At finite H I there is technically no phase transition, because no order parameter has a nonanalyticity in f a .)At high H I (low α ), this crossover is unimportant, because the potential is negligible anyway and the relaxation time(splitting) is simply the time for a state to spread across its domain. However, for β (cid:38) /α ( f a (cid:38) M P ( H I / Λ QCD )),the backreaction creates a strong enough potential to concentrate (cid:101) ρ around θ = π ; this is transPlanckian enough tobe practically irrelevant. The low- H I and high- H I behaviors both agree parametrically with the simple estimate ofthe backreaction transition given in the body of the paper. Appendix C: Classical Rolling Constraint
To prevent eternal inflation, we can impose the constraint that classical rolling dominates fluctuations for theinflaton itself, which leads to an upper bound on the length of inflation. The fluctuations are of order H ( t ) per e-fold,so we take the classical-rolling constraint to be, parametrically, ˙ φ (cid:38) H . More precisely,[76]˙ φ > (cid:114) π H (C1)With slow-roll, we obtain a lower bound on − ˙ V : ˙ φ ≈ − V (cid:48) H (C2) − ˙ V = − ˙ φV (cid:48) (C3) ≈ H ˙ φ (C4) > π H (C5)We can turn this into a lower bound on − ˙ H : − ˙ H = − ˙ V HM P (C6) > H π M P (C7)Note that this gives us a lower bound on the slow-roll parameter (cid:15) , (cid:15) = − ˙ H/H (C8) > H π M P (C9)8We can also invert it to obtain an upper bound on N : − dtdH < π M P H (C10) − dNdH < π M P H (C11) N < (cid:90) H i H f π M P H dH (C12)= 2 π M P H f − π M P H i (C13) < π M P H f (C14)where H f is the value of H when slow-roll inflation ends and H i > H f is the value when it begins.This bound agrees with the much more general result [70], which gives (in three spatial dimensions) that there is abound on the classical number of e-folds N c < S dS /
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