Initial conditions and sampling for multifield inflation
PPrepared for submission to JCAP
Initial conditions and sampling formultifield inflation
Richard Easther and Layne C. Price
Department of PhysicsUniversity of AucklandPrivate Bag 92019Auckland, New ZealandE-mail: [email protected], [email protected]
Abstract.
We investigate the initial conditions problem for multifield inflation. In thesescenarios the pre-inflationary dynamics can be chaotic, increasing the sensitivity of the onsetof inflation to the initial data even in the homogeneous limit. To analyze physically equivalentscenarios we compare initial conditions at fixed energy. This ensures that each trajectory iscounted once and only once, since the energy density decreases monotonically. We presenta full analysis of hybrid inflation that reveals a greater degree of long range order in theset of “successful” initial conditions than was previously apparent. In addition, we explorethe effective smoothing scale for the fractal set of successful initial conditions induced by thefinite duration of the pre-inflationary phase. The role of the prior information used to specifythe initial data is discussed in terms of Bayesian sampling.
Keywords: inflation, physics of the early universe, initial conditions, multifield, the measureproblem a r X i v : . [ a s t r o - ph . C O ] J u l ontents The standard hot big bang is synonymous with the Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) metric. This model imposes maximally symmetric initial conditions on the metricand the mass-energy distribution, as specified on an arbitrary initial spatial hypersurface.These initial conditions acausally correlate spacelike-separated regions and require furtherfine-tuning for the Universe to be spatially flat at late times, leading to the well-knownhorizon and flatness problems. Famously, these problems are resolved by inflation [1–4],which grafts a phase of accelerated expansion onto the very early universe, setting the stagefor the standard cosmology. During inflation the comoving Hubble volume contracts and thevisible universe is driven toward the spatially flat FLRW universe.Given that inflation attempts to explain the otherwise ad hoc initial conditions of thestandard hot big bang, a viable inflationary mechanism must itself be free of tunings. Tuningscan appear as technically unnatural parameter values in the inflaton sector or the need fora special pre-inflationary field configuration: the latter question is the focus of this paper.Inflationary models with unnatural initial conditions are at best incomplete and, at worst,not viable as descriptions of the early universe. Moreover, the level of tuning required toensure the onset of inflation can differ substantially between scenarios with largely degenerateobservational predictions, providing a possible mechanism for discriminating between them.The initial conditions problem arises even in the purely homogeneous limit. For instance,chaotic inflation [5] begins for a large range of initial field values, but new inflation withthe Coleman-Weinberg potential [3, 4] requires a special initial state. Inflationary modelswith multiple scalar degrees of freedom introduce a further level of complexity. With twoor more fields the homogeneous dynamics are potentially chaotic, as first pointed out inRef. [6] and also discussed by Refs [7, 8]. Chaos is synonymous with sensitive dependenceon initial conditions, rendering multifield models qualitatively different from their single fieldcounterparts. Multifield scenarios are widely studied and more natural in many settings.In particular, string theoretic inflationary scenarios often possess many scalar degrees offreedom. Further, even if a model has an effective single-field description once inflation isunderway, the pre-inflationary phase may contain many interacting fields. Several analysesof the initial conditions problem for multifield inflation exist [7–14] and we return to thisquestion here. – 1 –et Description I . . . . . . . Initial conditions surface, however defined Z . . . . . . . Set of initial conditions with zero velocity C E . . . . . . Set of initial conditions with equal energy E S E . . . . . . Successfully inflating subset of C E F E . . . . . . Non-inflating subset of C E B E . . . . . . Boundary between S E and F E Table 1 . Subsets of phase space: notation.
We need to sample the “initial conditions space” I for these scenarios, determining theoverall fraction that inflates and the topology of the inflationary region within this space. Ahomogeneous, spatially flat universe containing N scalar fields φ i with arbitrary interactionshas 2 N independent degrees of freedom since the scale factor can be eliminated by the 0-0Einstein equation. The solutions to the equations of motion — called trajectories or orbits— are non-intersecting curves that fill the 2 N –dimensional phase space. Yet, the initial fieldvalues and velocities are not independent or identically distributed (iid) random variables asdifferent points in I are correlated by the solutions to the field-equations, i.e. , many pointsbelong to the same trajectory. (See Table 1 for a summary of our notation.)The phase space is foliated by surfaces of equal energy, C E . The energy density ρ = E of FLRW universes is monotonic, decreasing in a homogeneous universe as˙ ρ ( t ) = − H N (cid:88) i =1 ˙ φ i , (1.1)where the Hubble parameter H ∝ E and overdots denote derivatives with respect to coor-dinate time t . For a specific energy E , orbits intersect C E once and only once, identifyingeach point on C E with a unique solution to the equations of motion. To build a well-definedsample of trajectories we choose initial conditions from the constraint surface C E .Many previous treatments of the multifield initial conditions problem [9–13] have beenbased on Z , the N –dimensional subset of I on which all velocities vanish simultaneously.Although the set of inflationary trajectories intersecting Z is easier to sample than C E , orbitsfor which all velocities vanish at the same instant are not generic, given the finite durationof the pre-inflationary era. Many orbits thus never intersect Z , while in principle others mayintersect it multiple times, correlating apparently distinct points — issues that cannot arisewhen sampling from C E . By contrast, Ref. [8] samples the full phase space I and also variesthe parameters in the potential itself, effectively marginalizing over the energy scale E . Thecurrent paper is the first analysis of the initial conditions problem for multifield inflation thatdoes not either (a) study a lower-dimensional surface in the initial conditions space (howeverdefined) or (b) sample the entirety of I , disregarding the fact that different points belong tothe same solution of the field equations. Any description of the primordial universe breaks down above some energy scale, andthis scale defines the appropriate initial conditions hypersurface C E for a well-specified Tetradis [11] presents a single equal energy slice, although a degree of freedom was removed by requiringthe velocities to be equal, similar to the projections we introduce for convenience in Figs 3–5. – 2 –odel. This energy will be associated with some scale in the particle physics sector, such asthe characteristic size of extra dimensions, the string scale, next-to-leading order correctionsto Einstein gravity, or ultimately its breakdown at the Planck scale. Points on C E are thusphysically commensurate, whereas points in I span several orders of magnitude in energy.Qualitatively, we will also find that the set of successfully inflating points has a simpler struc-ture and more obvious long range order when chosen from C E rather than Z , allowing us tobetter understand the underlying cosmological dynamics.Beyond the choice of initial conditions surface, we must also specify the prior probabilitydistributions (in Bayesian terms) for the initial field values and velocities. If ω is a probabilitydistribution that weights an initial condition x according to how well its final state matchesthe observed universe, then the expected value of ω over initial conditions x ∈ C E is (cid:104) ω ( C E ) (cid:105) = (cid:90) C E ω ( x ) P E ( x ) d N x ≈ n n (cid:88) i =1 (cid:110) ω ( x ( i )0 ) (cid:111) x ( i )0 ∈ C E , (1.2)where P E is the prior probability distribution for initial conditions on the constraint surface C E and the sum is evaluated at n points sampled from C E . The prior in Eq. (1.2) actsas the probability density function for initial conditions on the space of FLRW universes.The form of P E is only weakly constrained by fundamental considerations. The freedom tochoose P E is analogous to the measure problem in the multiverse [15–19], albeit restrictedto the subspace of homogeneous FLRW universes. In many previous works the prior is oftennot directly discussed, and thus implicitly defined as a uniform distribution on the initialconditions. We consider several possible choices of prior (all of which are uninformative)and vary the energy of the surfaces C E . We find that the choice of prior significantly altersthe fraction of trajectories that lead to inflation, potentially distorting conclusions about theextent to which a given inflationary model requires fine-tuned initial conditions.In what follows we work with a widely studied two-field model: canonical hybrid or false-vacuum inflation [20–22]. We relate the initial conditions problem to that of determining the(fractal) topology and the geometry of the subset of points S E ⊂ C E that successfully inflate,since this is independent of the choice of prior. Like Refs [8, 13] we see that S E has a fractaltopology due to the presence of chaos in the underlying dynamical system, demonstratingthat hybrid inflation has regions of phase space where orbits are highly sensitive to theirinitial conditions and confirming the results of Ref. [6]. Hybrid inflation is associated with ablue power spectrum at odds with recent astrophysical data [25–29]. However, our primaryfocus is not hybrid inflation itself, but developing tools that can be used to understandthe initial conditions problems in generic models of multifield inflation. We use this modelbecause (a) it is the prototypical multifield model with chaotic dynamics and a narrowlydefined inflationary attractor; (b) we are primarily interested in the onset of inflation; and(c) to make contact with previous work.The pre-inflationary universe is dissipative, so the fractal structure must have a nontriv-ial scale dependence: there is necessarily a minimum scale below which two nearby trajecto-ries will remain correlated until they reach either the inflationary attractor or a minimum ofthe potential, smoothing S E below this scale. Conversely, while we assume classical homo-geneity, quantum fluctuations prevent the universe from being perfectly smooth. If S E has If the potential has one or more local minima where V Λ >
0, choosing E < V Λ will necessarily excludeall trajectories which evolve toward these minima. Although see Refs [23, 24]. – 3 –tructure on scales smaller than a typical fluctuation we cannot sensibly define the homoge-neous limit for this system. Consequently, we propose a sampling technique that identifiesregions where S E has structure below this minimum scale.The paper is arranged as follows: in Section 2 we review hybrid inflation and discussits dynamics. In Section 3 we describe our numerical methods, characterize the properties ofthe set of inflationary trajectories with different energies and priors P E , and investigate thefractal dimension of S E . In Section 4 we discuss the implication of our results and identifyfuture lines of enquiry. For simplicity we consider two homogeneous scalar fields, ψ and the inflaton φ , interactingthrough a potential V ( ψ, φ ) in a homogeneous FLRW universe. The equations of motion are¨ φ + 3 H ˙ φ + ∂V∂φ = 0 and ¨ ψ + 3 H ˙ ψ + ∂V∂ψ = 0 , (2.1)and the Hubble parameter H can be eliminated by the 0-0 Einstein equation H = 8 π M (cid:20)
12 ˙ φ + 12 ˙ ψ + V ( ψ, φ ) (cid:21) , (2.2)where M Pl is the Planck mass. Following Refs [6–13] we consider hybrid inflation [20–22]with the potential V ( ψ, φ ) = Λ (cid:34)(cid:18) − ψ M (cid:19) + φ µ + φ ψ ν (cid:35) , (2.3)with real parameters Λ, M , µ , and ν . Inflation occurs in the “inflationary valley” with ψ ≈ | φ | > φ c , where φ c = √ ν /M is the critical point at which the effective massof ψ becomes complex and inflation comes to an end. The potential is symmetric under φ → − φ and ψ → − ψ , with two equivalent valleys for φ > φ c and φ < − φ c and minima at { ψ, φ } = {± M, } . Orbits will either enter one of the false-vacuum inflationary valleys orevolve directly toward one of the true vacua.We set the amplitude A s of the dimensionless power spectrum P R to be roughly com-patible with the WMAP9 data [26, 27], which fixes the potential energy scale. This resultsin A s ≈ π M (cid:18) V(cid:15) V (cid:19) = (2 . ± . × − , (2.4)where (cid:15) V = ( M / V ,φ /V ) is the slow-roll parameter. Setting M = . M Pl , µ = 500 M Pl ,and ν = . M Pl and assuming perturbations are generated when ψ ≈ φ ≈ φ c , wederive Λ ≈ . × − M Pl . Lastly, Ref. [30] determined that quantum fluctuations dominatethe classical field evolution in the inflationary valley whenΛ > Λ q ≡ π √ M φ c µ . (2.5)For our parameters Λ q = 9 . × − M Pl , so the classical equations are self-consistent. The super-Planckian value of µ is an artifact of this definition of the potential; the actual mass term is m φ = 2Λ /µ ≈ − M , and safely sub-Planckian. – 4 – .05 .1 .15 .20.05.1.15.2 Ψ (cid:144) M Pl Φ (cid:144) M P l Figure 1 . Distribution of successfully inflating initial conditions drawn from the zero-velocity slice Z . White areas have the highest number of successful points; darker regions have the fewest. Thisplot matches Fig. (1) of Ref. [8], with M = . M Pl , µ = 636 M Pl , and ν = . M Pl . The figuresare similar (verifying our codes and algorithms) but are not expected to be identical, due to differentbinning procedures. The inflationary valley is a small subset of the total phase space, which might suggestthe model has a fine-tuning problem. References [9, 11, 12] considered sub-Planckian initialfield values on the zero-velocity surface Z , pessimistically concluding that — in the absenceof effects that increase the friction experienced by the fields — only trajectories which startinside the inflationary valley yield 60 e-folds of inflation. By contrast, Ref. [10] was more op-timistic, showing that a supergravity-inspired hybrid inflation model has a significant numberof “successful” points outside the inflationary valley. With more exhaustive sampling of Z ,subsequent studies by Clesse, Ringeval, and Rocher [8, 13] extended this optimistic conclusionto the potential (2.3). They showed that successful initial conditions are distributed in anintricate series of patches and fine lines outside the inflationary valley, with a fractal bound-ary separating inflating and non-inflating initial conditions. The distribution of successfullyinflating initial conditions on Z , for a specific scenario from Ref. [8], is reproduced in Fig. 1.The fine-tuning problem may also be less serious if the initial field values are assumed to besuper-Planckian or if the interaction term dominates [13, 31].References [8, 23] also present a Markov Chain Monte Carlo (MCMC) sampling ofall possible parameter choices and sub-Planckian field configurations, including those with– 5 –nitial velocities. The conclusion was that 60 e-folds of inflation is generic for the potentialin Eq. (2.3) and fine-tuned initial conditions in the inflationary valley are not required.Although we have argued that sampling from any two-dimensional subspace, such as Z , is oflimited benefit, sampling the whole four-dimensional space I may not be strictly necessary,even though an MCMC technique marginalizes the unknown initial energy E . We insteadchoose to explore how fine-tuned the initial conditions must be when sampling from constraintsurfaces C E that incorporate the energy constraint Eq. (1.1). We numerically integrate Eqs (2.1)–(2.3) using a backward-difference formula implemented bythe
Fcvode package from the
Sundials computing suite [32]. We sample initial conditionsfrom constraint surfaces C E with constant energy density ρ = 12 ˙ ψ + 12 ˙ φ + V ( ψ , φ ) = E , (3.1)where E = 10 i M Pl for i ∈ {− , . . . , } . (3.2)The last 60 e-folds of inflation occur at { ψ, φ } ≈ { , φ c } with E ∼ − M Pl . With E =10 M Pl we are at the limit of classical Einstein gravity; we only include this case to illustratethe underlying dynamical system.We stop integrating when either (a) the orbit achieves more than 60 e-folds duringinflation or (b) ρ < Λ and the trajectory is trapped by the potential wells at { ψ, φ } = {± M, } . Initial conditions which lead to 60 e-folds of inflation are “successful” and definethe subset S E , while its complement — the “failed” points — comprise the subset F E . Theboundary between these sets, whose properties determine the extent to which they “mix,” isdenoted B E .We select points randomly on the constraint surface as follows. We first draw φ and ψ from the uniform distribution over 0 ≤ { ψ , φ } ≤ . M Pl , excluding any choices with V ( φ , ψ ) > E . The symmetry of the potential (2.3) allows the restriction to positive fieldvalues, whereas the upper bound is set to be consistent with Ref. [8]. Given these initial fieldvalues, the kinetic energy is typically dominant unless E ≈ Λ ∼ − M Pl . We apportion theremaining energy by drawing one of v ∈ { ˙ φ , ˙ ψ } from a uniform prior on the range − (cid:112) E − V ) ≤ v ≤ (cid:112) E − V ) (3.3)and giving the leftover energy to the other field velocity v with the overall sign again chosenrandomly. We maintain the symmetry between ˙ φ and ˙ ψ by alternating the order in whichthese velocity terms are set. This procedure implicitly defines the initial priors P orig on theconstraint surfaces C E .To estimate the size of inhomogeneous fluctuations at the initial energy E , we note that δφ ∼ H/ π and H ∼ E /M Pl for a massless field in de Sitter space. Similarly, the minimal Again, additional constraints can be added, e.g. data matching for n s , r , or other observables. This prior generates the high tails in the velocity distributions seen in Fig. 6. If the first velocity chosenis v , the second will be v = ± (cid:112) E − V ) − v . Since v is uniformly distributed, v is a quadraticdistribution, favoring higher values. – 6 –ariation in velocities is expected to be of order H across a Hubble volume [30, 33, 34].The fields ψ and φ are not massless and the pre-inflationary universe is not de Sitter, butwe can use this relationship to put an approximate lower bound on the homogeneity of theprimordial universe. In regions of S E whose typical scale in any phase space dimension isless than ∆ ≡ { δψ , δφ , δ ˙ ψ , δ ˙ φ } = 12 π { H, H, H , H } (3.4)the homogeneous approximation breaks down and further analysis is invalid or ambiguous.Physically, in these regions we cannot self-consistently assume that the primordial universeis homogeneous.We exclude these regions from S E by sampling C E in clusters. We first choose pointsfrom C E and integrate Eqs (2.1)–(2.3). At each point that successfully inflates we randomlydraw 100 points within ∆ of that point. If any of these new points do not inflate, we concludethat the original point was a “false” (or perhaps ambiguous) positive. Applying this simplestability check at various other points along the trajectory’s evolution is straightforward,but computationally expensive. Furthermore, the largest fluctuations occur at the highestenergies, so testing the initial energy surface captures the most relevant effects. Although thisapproach incorporates points lying near (but not actually on) our designated equal energysurface, we do not weight our conclusions by these secondary points.This analysis does not address the inhomogeneous initial conditions problem; it sim-ply limits the extent to which the initial conditions can be self-consistently fine-tuned in ahomogeneous universe, given that the chaotic dynamics of the potential may cause closelycorrelated trajectories to diverge exponentially. Fig. 2 shows three solutions of Eqs (2.1)–(2.3), at E = 10 − M Pl with initial field values which differ by only 10 − M Pl . They eventuallydiverge, with each trajectory reaching a distinct end-state. If an inflation model has a frac-tal S E or B E that is distributed in a complex manner over C E , then almost all successfullyinflating initial conditions may be within ∆ of an initial condition which does not inflate. The fraction of successful points at any given energy E is summarized in Table 2, bothincluding and excluding the “false positives.” The highest probability for success is at higherenergies. We should expect this since, given that the effective equation of state is not thesame on all trajectories, orbits accumulate on the narrow inflationary attractor over time,leaving a larger flux of orbits through the attractor at lower energies. In comparison to asample drawn from C E , an identical sample from a slice C E (cid:48) with E > E (cid:48) will not placeas much weight on trajectories inside the inflationary attractor and we expect to see fewersuccessfully inflating initial conditions on the lower energy surface.Figures 3 to 5 show two dimensional slices of S E at different values of E . We bin S E on a 1000 × E = 10 − M Pl ), and 5 ( E = 10 − M Pl ) show two-dimensional slices of C E on which theinitial velocities have equal magnitude | ˙ φ | = | ˙ ψ | . Looking at Figs 3 through 5 we cansee areas where S E and F E mix together, forming an intricate substructure similar to that For the lowest values of E , (cid:110) δψ /ψ max , δφ /φ max , δ ˙ ψ / ˙ ψ max , δ ˙ φ / ˙ φ max (cid:111) ∼ − , which is far below theresolution of our figures. We confirmed the accuracy of the Fcvode integrator in this domain by using anarbitrary precision integrator from Mathematica. – 7 – Ψ (cid:144) M Pl (cid:45) Φ (cid:144) M Pl t (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ψ (cid:144) M Pl Φ (cid:144) M P l Figure 2 . Parametric trajectory plots for three orbits at energy E = 10 − M Pl initially separatedby 10 − M Pl in the field values and 10 − M in the field velocities. These orbits are exponentiallydiverging, with each of the three trajectories branching at t ≈ M − ; one goes to each of theglobal minima at ψ = ± M and one inflates. E [ M Pl ] n succ [M] n total [M] n false [k] f true f total − − − − † − Table 2 . Total fraction of successfully inflating points sampled from priors P orig on the equal energyslices C E — both excluding ( f true ) and including ( f total ) false positives from S E . Also shown are thenumber of successful points n succ , the number of false positives n false , and the combined number offail points, false positives, and successful points n total . The numbers n succ and n total are measured inmillions [M] of points, n false is measured in thousands [k] of points, and the energy E is in units ofthe Planck mass M Pl . ( † ) The sampling procedure deviates from an “equal-area” sample as E → Λ. seen in Refs [8, 13]. In Fig. 4 we also see contiguous regions and thick bands which reliablyinflate and survive the subtraction of the false positives from S E . Qualitatively, S E exhibitsconsiderable long range order when compared to Fig. 1. At higher energy, contiguous regionsin S E occupy a larger portion of C E , which can be seen clearly in Figs 3 and 4. The transitionfrom Figs 3 to 5 shows how the geometry of S E changes with E .Figures 3 to 5 are projections of three dimensional regions and suppress informationabout the field velocities of the successful initial configurations. Intuitively, the points mostlikely to inflate (for given φ and ψ ) would be those which had a large | ˙ φ | and small | ˙ ψ | .These points are essentially “launched” up the inflationary valley, while the slope of thepotential focuses them toward smaller values of ψ . To show this dependence on the initialvalues ˙ φ and ˙ ψ , Fig. 6 shows histograms of these values sampled from the whole of C E . The– 8 – .05 .1 .15 .20.05.1.15.2 Ψ (cid:144) M Pl Φ (cid:144) M P l Ψ (cid:144) M Pl Φ (cid:144) M P l Ψ (cid:144) M Pl Φ (cid:144) M P l Ψ (cid:144) M Pl Φ (cid:144) M P l Figure 3 . Two dimensional slicings of C E for E = 10 − M Pl , including the ambiguous, “false positive”points in S E . Parameters are Λ = 6 . × − M Pl , M = . M Pl , µ = 500 M Pl , and ν = . M Pl . Thelight and dark areas are regions that have a higher and lower density of points in S E , respectively.The results have been binned over a 1000 × φ >
0, the right column is at ˙ φ <
0, the top row has ˙ ψ >
0, and the bottom rowhas ˙ ψ < fraction of sampled points in S E as a function of initial velocity confirms our intuition: mostsuccessful points have larger | ˙ φ | and smaller | ˙ ψ | . Points for which ˙ φ ≈ Z is unrepresentative of typicalinflationary trajectories. We can also see the impact of the “false positives” in these plots:these are more frequent at high energies and in the limiting case E = M Pl all na¨ıvely-inflatinginitial conditions are false positives, since ∆ encompasses the whole of I in this limit.With sub-Planckian initial field values the kinetic energy dominates the potential energyfor E (cid:29) Λ. Thus, even if a trajectory starts inside the inflationary valley, its velocity is such– 9 – .05 .1 .15 .20.05.1.15.2 Ψ (cid:144) M Pl Φ (cid:144) M P l Ψ (cid:144) M Pl Φ (cid:144) M P l Figure 4 . Two dimensional slicings of C E for E = 10 − M Pl , excluding any ambiguous or “false” pos-itives from the set of successfully inflating initial conditions, S E . All velocities are of equal magnitude;the left panel has ˙ ψ < φ >
0; and the right panel has { ˙ ψ , ˙ φ } < Ψ (cid:144) M Pl Φ (cid:144) M P l Ψ (cid:144) M Pl Φ (cid:144) M P l Figure 5 . Two dimensional slicings of C E for E = 10 − M Pl , where the checked background has V ( ψ , φ ) > E and has not been sampled. All velocities are of equal magnitude; the left panel has { ˙ ψ , ˙ φ } > { ˙ ψ , ˙ φ } < that it is unlikely to remain there. For example, when E = 10 − M Pl the slices of S E in Figs 3 and 4 show no special preference for points within the valley. In contrast, with E = 10 − M Pl ∼
10Λ the valley is clearly distinguishable, as shown in Fig. 5, but only whenthe initial velocity of φ is directed “uphill,” i.e. with ˙ φ ≥
0. Conversely, each slice containsmany successful points that lie outside the inflationary valley.In Fig. 7 we project specific representative solutions of Eqs (2.1)–(2.3) onto the { ψ, φ } – 10 – (cid:45) Φ(cid:160) (cid:144) M Pl2 (cid:45) (cid:45)
Ψ(cid:160) (cid:144) M Pl2 (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Φ(cid:160) (cid:144) M Pl2 (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Ψ(cid:160) (cid:144) M Pl2 (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Φ(cid:160) (cid:144) M Pl2 (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Ψ(cid:160) (cid:144) M Pl2
Figure 6 . Histograms of ˙ φ (left column) and ˙ ψ (right column) drawn from equal energy surfaces C E with priors P orig . The rows have energy 10 − M Pl (top), 10 − M Pl (middle), and 10 − M Pl (bottom).The gray background is the total sample from C E and the blue foreground is the successful subset S E .At E = M Pl (not displayed), there are false positives only. plane for initial conditions with energies E = 10 − M Pl and E = 10 − M Pl . Trajectorieswhich unambiguously inflate show little topological mixing and are all reflected off of themaximum of the potential V max = V | ψ = φ toward the inflationary valley. For E (cid:38) − M Pl ,most trajectories contain regions in which the field values are super-Planckian. We do notexclude these trajectories, but we could easily add this as a separate requirement for a viableinflationary scenario, in which case almost no successful inflationary trajectories exist at these– 11 – (cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:45) (cid:45) (cid:45) (cid:45) Ψ (cid:144) M Pl Φ (cid:144) M P l (cid:230) (cid:230)(cid:230)(cid:230) (cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:45) .15 (cid:45) .05 (cid:45) .1 0 .1.05 .15 (cid:45) (cid:45) Ψ (cid:144) M Pl Φ (cid:144) M P l Figure 7 . Successfully inflating trajectories projected onto the { ψ, φ } plane, with initial conditions { ψ , φ } marked in red. The left panel is at energy E = 10 − M Pl and the right panel is at energy E = 10 − M Pl . The gray, checked region is where the magnitude of the field values exceeds M Pl . energies.To quantify the sensitive dependence on initial conditions independently of our samplingprocedure, we use the box-counting method to estimate the fractal dimension of both S E andits boundary B E , including the “false” positives [35]. We first cover each set S E and B E withprogressively smaller four-dimensional boxes of size δ , then count the number N ( δ ) of δ -sizedboxes in each covering. The box-counting dimension d = lim δ → log( N ( δ ))log(1 /δ ) (3.5)is estimated by the slope of the line fitted to the linear portion of the curve log( N ) as afunction of log(1 /δ ). To compute the dimension of B E we count boxes that contain elementsof both S E and F E . Figure 8 shows both a typical fit (for B E with E = 10 − M Pl ) and thecomputed values of d for S E and B E . The result is sensitive to the detailed fitting procedure,which we trained by testing the algorithm on sets with known dimension, such as Cantordust. Furthermore, the estimate for d depends both on the non-trivial distribution of S E over C E and the resolution of sampled points, which is a function of the initial energy andsampling prior. The reported values of d should be interpreted as an upper bound to themore fundamental Hausdorff dimension [35] that improves with increasing E , where the set S E has higher long-range order.The regions considered here are multifractal, in that the dimension of S E will be afunction of both position in I and the overall scale. The first is easy to see: each surface C E contains regions in which essentially all points inflate (in these regions d ≈
3) and regionsthat are approximately isolated points (with d < d is effectively a weighted average of at least two different regions, which explains why thedimension of S E is close to 3 but still measurably non-integer. Secondly, at very small scales S E must consist of smooth contiguous regions and on these scales we expect d →
3. These– 12 – (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) log (cid:72) ∆ (cid:76) l og (cid:72) N (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) log (cid:72) E (cid:76) d Figure 8 . The left panel shows the box counting of B E for E = 10 − M Pl . The slope of the best-fitline d = 2 .
558 is the box counting dimension. The right panel is the box counting dimension d versusthe energy E for sets S E and B E at the energies in Eq. (3.2). The red line with boxes indicates S E and the blue line with circles is B E . — P orig P square P ˙ φ P ˙ ψ E [ M Pl ] f true f total f true f total f true f total f true f total − − − − − Table 3 . Fraction f of sampled points from C E that inflate — both excluding (true) and including(total) false positives. The sampling techniques ( P orig , P square , P ˙ φ , and P ˙ ψ ) are explained in the text. regions exist in spite of the chaotic dynamics due to the dissipative terms in Eqs (2.1)–(2.3)and put a lower limit on the mixing scale. It is well-known [15–19, 31, 36] that probability measures on different hypersurfaces result indifferent conclusions regarding the likelihood of inflation; we explore here how this relates tothe choice of sampling prior. Although surfaces with different energies (as well as differentinitial conditions surfaces, such as a slice of constant comoving time) are homeomorphic to C E and, by definition, have the same topology, the prior on C E is not a topological propertyand is not preserved under either homeomorphism or a change of variables. Each initialcondition surface then has a different prior and different likelihood for inflation, even giventhe same sampling technique.In Table 3 we compare different uninformative priors, defined implicitly through foursampling algorithms, on surfaces C E at the energies in Eq. (3.2). Since the kinetic en-ergy is initially dominant for the energies and ranges we consider, we leave the selection A homeomorphism is provided by time-translation along the integral curves of the equations of motion.Note that Z , being of a lower dimension, is not homeomorphic to C E . – 13 –ethod for { ψ , φ } the same as in Section 3.1, but vary the way we set the velocities v ∈ { ˙ ψ , ˙ φ } . The original prior P orig draws one velocity v from a uniform distribution,bounded by ± (cid:112) E − V ), and then sets v by the energy constraint (3.1). All signs arechosen randomly and this procedure is alternated on subsequent choices of points to obtain asymmetric distribution in the velocities. The second prior P square is similar, except we drawthe square of the velocity v from a uniform distribution bounded below by zero and aboveby 2( E − V ), with the sign of v chosen randomly. Again, we alternate this to obtain asymmetric distribution. With this modest change in prior, the calculated fraction f true of C E that inflates (excluding false positives) differs by only a few percent, with P square giving aslightly lower fraction at each energy. The fraction f true again decreases with decreasing E .We compare this to two priors P ˙ φ and P ˙ ψ that are asymmetric in the velocities. For P ˙ φ we always draw ˙ ψ from a uniform distribution bounded by ± (cid:112) E − V ) and alwaysset ˙ φ by the energy constraint. For P ˙ ψ we do the opposite: draw ˙ φ and set ˙ ψ . Thisgives a uniform distribution in the sampled velocity v , but a high-tail distribution similar toFig. 6 in the velocity v . The prior P ˙ φ focuses more of the sample around ˙ ψ ≈
0, the areaidentified as being most-likely to inflate, and P ˙ ψ gives more points around ˙ φ ≈
0, the arealeast-likely to inflate. At E = M Pl the difference in f total (including false positives) betweenthe asymmetric priors is as much as 41.0 percentage points. The differences decrease withdecreasing E , indicating that later-time hypersurfaces become progressively independent ofthe prior. However, Table 3 demonstrates how any measure of f is prior-dependent, especiallywith respect to the implicit dependence of the prior on the initial energy. We have considered the initial conditions problem for multifield inflation, quantifying thelikelihood of inflation by sampling an initial conditions surface, evolving the points numeri-cally, and dividing them into successfully and unsuccessfully inflating sets. We draw initialconditions from an equal energy slice of phase space, denoted C E , the maximum energy atwhich the underlying theory is assumed to be an accurate description of the primordial uni-verse. Since FLRW universes have a monotonic energy density, sampling initial conditionsfrom C E ensures that we count only unique solutions to the equations of motion. A sample ofpoints from C E is thus a well-defined sample of homogeneous universes. Typically, we cannotpredict the flux of orbits through C E and must choose a prior, accordingly. We consideredfour different uninformative priors on C E and showed that the likelihood of inflation variesby as much as a factor of roughly two between candidate priors. However, one can imaginescenarios where the prior dependence was much more dramatic.After specializing to hybrid inflation we examined the topology of the set of successfulpoints S E , which is independent of continuous deformations to the prior. We confirm thatboth S E and the boundary between the successful and unsuccessful points is fractal for allsampled energies. The structure of S E , as seen in Figs 3 to 5, is qualitatively smootherthan when initial conditions are chosen from the zero-velocity slice shown in Fig. 1. Further,since the equations of motion (2.1)–(2.3) are dissipative, there must be a small-scale cutoffto any fractal structure. However, quantum fluctuations put a fundamental lower limit onthe homogeneity of the early universe: if S E has structure below this scale, the assumptionof homogeneity is not self-consistent. Fluctuations are larger at higher energies and abovesome critical energy E the number of viable, homogeneous scenarios is vanishingly small, even– 14 –hough the na¨ıve counting statistic suggests that a nontrivial fraction of the initial conditionsspace is inflationary.Our specific calculations are performed for the hybrid potential (2.3), but our underlyinggoal is to develop tools that can be applied to the initial conditions problem associatedwith generic multifield scenarios. Recent progress has been made by studying both randommultifield models [37–42] and inflection point models [36, 42–46]. These approaches yieldcontrasting conclusions regarding the distribution of inflationary trajectories; applying themethods developed here to these models will be an interesting extension of this work.This analysis assumes that the universe is initially spatially flat and homogeneous, buteven if inflation begins without tuning in the homogeneous limit there is no guarantee thatthis result will survive the addition of pre-inflationary inhomogeneities. Inhomogeneous pre-inflationary configurations were examined by Goldwirth and Piran [47–49], who showed thatsingle-field chaotic inflation and new inflation [3, 4] remain robust in the presence of nontrivialinhomogeneity, provided that the initial field value is approximately correlated over severalHubble radii. We plan to examine this question for multifield inflation in future work. Acknowledgments
The authors acknowledge the contribution of the NeSI high-performance computing facilitiesand the staff at the Centre for eResearch at the University of Auckland. New Zealand’s na-tional facilities are provided by the New Zealand eScience Infrastructure (NeSI) and fundedjointly by NeSI’s collaborator institutions and through the Ministry of Science & Innovation’sResearch Infrastructure programme [ ]. We thank the Yukawa In-stitute for Theoretical Physics at Kyoto University, where a draft of this work was presentedat workshop YITP-T-12-03, and Grigor Aslanyan, S´ebastien Clesse, and Christophe Ringevalfor helpful comments on the manuscript. We also thank the Center for Applied ScientificComputing at LLNL for making the
Sundials package freely available.
References [1] A. A. Starobinsky,
A New Type of Isotropic Cosmological Models Without Singularity , Phys.Lett.
B91 (1980) 99–102.[2] A. H. Guth,
The Inflationary Universe: A Possible Solution to the Horizon and FlatnessProblems , Phys.Rev.
D23 (1981) 347–356.[3] A. D. Linde,
A New Inflationary Universe Scenario: A Possible Solution of the Horizon,Flatness, Homogeneity, Isotropy and Primordial Monopole Problems , Phys.Lett.
B108 (1982)389–393.[4] A. Albrecht and P. J. Steinhardt,
Cosmology for Grand Unified Theories with RadiativelyInduced Symmetry Breaking , Phys.Rev.Lett. (1982) 1220–1223.[5] A. D. Linde, Chaotic Inflation , Phys.Lett.
B129 (1983) 177–181.[6] R. Easther and K.-i. Maeda,
Chaotic dynamics and two field inflation , Class.Quant.Grav. (1999) 1637–1652, [ gr-qc/9711035 ].[7] R. O. Ramos, Fine tuning solution for hybrid inflation in dissipative chaotic dynamics , Phys.Rev.
D64 (2001) 123510, [ astro-ph/0104379 ].[8] S. Clesse, C. Ringeval, and J. Rocher,
Fractal initial conditions and natural parameter values inhybrid inflation , Phys.Rev.
D80 (2009) 123534, [ arXiv:0909.0402 ]. – 15 –
9] G. Lazarides, C. Panagiotakopoulos, and N. Vlachos,
Initial conditions for smooth hybridinflation , Phys.Rev.
D54 (1996) 1369–1373, [ hep-ph/9606297 ].[10] G. Lazarides and N. Vlachos,
Initial conditions for supersymmetric inflation , Phys.Rev.
D56 (1997) 4562–4567, [ hep-ph/9707296 ].[11] N. Tetradis,
Fine tuning of the initial conditions for hybrid inflation , Phys.Rev.
D57 (1998)5997–6002, [ astro-ph/9707214 ].[12] L. E. Mendes and A. R. Liddle,
Initial conditions for hybrid inflation , Phys.Rev.
D62 (2000)103511, [ astro-ph/0006020 ].[13] S. Clesse and J. Rocher,
Avoiding the blue spectrum and the fine-tuning of initial conditions inhybrid inflation , Phys.Rev.
D79 (2009) 103507, [ arXiv:0809.4355 ].[14] N. Agarwal, R. Bean, L. McAllister, and G. Xu,
Universality in D-brane inflation , Journal ofCosmology and Astroparticle Physics (2011), no. 09 002, [ arXiv:1103.2775 ].[15] G. Gibbons, S. W. Hawking, and J. Stewart,
A natural measure on the set of all universes , Nuclear Physics B (1987), no. 3 736–751.[16] S. Hawking and D. N. Page,
How probable is inflation? , Nucl.Phys.
B298 (1988) 789–809.[17] G. Gibbons and N. Turok,
The Measure Problem in Cosmology , Phys.Rev.
D77 (2008) 063516,[ hep-th/0609095 ].[18] B. Freivogel,
Making predictions in the multiverse , Class.Quant.Grav. (2011) 204007,[ arXiv:1105.0244 ].[19] J. S. Schiffrin and R. M. Wald, Measure and Probability in Cosmology , Phys.Rev.
D86 (2012)023521, [ arXiv:1202.1818 ].[20] L. Kofman and A. D. Linde,
Generation of Density Perturbations in the InflationaryCosmology , Nucl.Phys.
B282 (1987) 555.[21] A. D. Linde,
Hybrid inflation , Phys.Rev.
D49 (1994) 748–754, [ astro-ph/9307002 ].[22] E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart, and D. Wands,
False vacuuminflation with Einstein gravity , Phys.Rev.
D49 (1994) 6410–6433, [ astro-ph/9401011 ].[23] S. Clesse,
Hybrid inflation along waterfall trajectories , Phys.Rev.
D83 (2011) 063518,[ arXiv:1006.4522 ].[24] H. Kodama, K. Kohri, and K. Nakayama,
On the waterfall behavior in hybrid inflation , Prog.Theor.Phys. (2011) 331–350, [ arXiv:1102.5612 ].[25] K. Story, C. Reichardt, Z. Hou, R. Keisler, K. Aird, et al.,
A Measurement of the CosmicMicrowave Background Damping Tail from the 2500-square-degree SPT-SZ survey , arXiv:1210.7231 .[26] G. Hinshaw, D. Larson, E. Komatsu, D. Spergel, C. Bennett, et al., Nine-Year WilkinsonMicrowave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results , arXiv:1212.5226 .[27] C. Bennett, D. Larson, J. Weiland, N. Jarosik, G. Hinshaw, et al., Nine-Year WilkinsonMicrowave Anisotropy Probe (WMAP) Observations: Final Maps and Results , arXiv:1212.5225 .[28] J. L. Sievers, R. A. Hlozek, M. R. Nolta, V. Acquaviva, G. E. Addison, et al., The AtacamaCosmology Telescope: Cosmological parameters from three seasons of data , arXiv:1301.0824 .[29] Planck Collaboration
Collaboration, P. Ade et al.,
Planck 2013 results. I. Overview ofproducts and scientific results , arXiv:1303.5062 .[30] J. Martin and V. Vennin, Stochastic Effects in Hybrid Inflation , Phys.Rev.
D85 (2012) 043525,[ arXiv:1110.2070 ]. – 16 –
31] G. N. Felder, L. Kofman, and A. D. Linde,
Inflation and preheating in NO models , Phys.Rev.
D60 (1999) 103505, [ hep-ph/9903350 ].[32] A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S.Woodward,
Sundials: Suite of nonlinear and differential/algebraic equation solvers , ACMTrans. Math. Softw. (Sept., 2005) 363–396.[33] F. Finelli, G. Marozzi, A. Starobinsky, G. Vacca, and G. Venturi, Stochastic growth of quantumfluctuations during slow-roll inflation , Phys.Rev.
D82 (2010) 064020, [ arXiv:1003.1327 ].[34] F. Finelli, G. Marozzi, A. Starobinsky, G. Vacca, and G. Venturi,
Generation of fluctuationsduring inflation: Comparison of stochastic and field-theoretic approaches , Phys.Rev.
D79 (2009) 044007, [ arXiv:0808.1786 ].[35] J. Theiler,
Estimating fractal dimension , J. Opt. Soc. Am. A (1990) 1055–1073.[36] S. Downes, B. Dutta, and K. Sinha, Attractors, Universality and Inflation , Phys.Rev.
D86 (2012) 103509, [ arXiv:1203.6892 ].[37] A. Aazami and R. Easther,
Cosmology from random multifield potentials , JCAP (2006)013, [ hep-th/0512050 ].[38] S.-H. H. Tye, J. Xu, and Y. Zhang,
Multi-field Inflation with a Random Potential , JCAP (2009) 018, [ arXiv:0812.1944 ].[39] J. Frazer and A. R. Liddle,
Multi-field inflation with random potentials: field dimension,feature scale and non-Gaussianity , JCAP (2012) 039, [ arXiv:1111.6646 ].[40] D. Marsh, L. McAllister, and T. Wrase,
The Wasteland of Random Supergravities , JHEP (2012) 102, [ arXiv:1112.3034 ].[41] D. Battefeld, T. Battefeld, and S. Schulz,
On the Unlikeliness of Multi-Field Inflation:Bounded Random Potentials and our Vacuum , JCAP (2012) 034, [ arXiv:1203.3941 ].[42] L. McAllister, S. Renaux-Petel, and G. Xu,
A Statistical Approach to Multifield Inflation:Many-field Perturbations Beyond Slow Roll , JCAP (2012) 046, [ arXiv:1207.0317 ].[43] N. Itzhaki and E. D. Kovetz,
Inflection Point Inflation and Time Dependent Potentials inString Theory , JHEP (2007) 054, [ arXiv:0708.2798 ].[44] R. Allahverdi, B. Dutta, and A. Mazumdar,
Attraction towards an inflection point inflation , Phys.Rev.
D78 (2008) 063507, [ arXiv:0806.4557 ].[45] N. Itzhaki and E. D. Kovetz,
A Phase Transition between Small and Large Field Models ofInflation , Class.Quant.Grav. (2009) 135007, [ arXiv:0810.4299 ].[46] M. Spalinski, Initial conditions for small field inflation , Phys.Rev.
D80 (2009) 063529,[ arXiv:0903.4999 ].[47] D. S. Goldwirth and T. Piran,
Inhomogeneity and the onset of inflation , Phys.Rev.Lett. (1990) 2852–2855.[48] D. S. Goldwirth, On inhomogeneous initial conditions for inflation , Phys.Rev.
D43 (1991)3204–3213.[49] D. S. Goldwirth and T. Piran,
Initial conditions for inflation , Phys.Rept. (1992) 223–291.(1992) 223–291.