Scalar correlation functions for a double-well potential in de Sitter space
IIMPERIAL-TP-2020-AR-1
Scalar correlation functions for adouble-well potential in de Sitterspace
Tommi Markkanen a,b and Arttu Rajantie c a Laboratory of High Energy and Computational Physics, National Institute of ChemicalPhysics and Biophysics, R¨avala pst. 10, Tallinn, 10143, Estonia b Helsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland c Department of Physics, Imperial College London, London, SW7 2AZ, United KingdomE-mail: tommi.markkanen@kbfi.ee, [email protected]
Abstract.
We use the spectral representation of the stochastic Starobinsky-Yokoyama ap-proach to compute correlation functions in de Sitter space for a scalar field with a symmetricor asymmetric double-well potential. The terms in the spectral expansion are determined bythe eigenvalues and eigenfunctions of the time-independent Fokker-Planck differential oper-ator, and we solve them numerically. The long-distance asymptotic behaviour is given bythe lowest state in the spectrum, but we demonstrate that the magnitude of the coeffients ofdifferent terms can be very different, and the correlator can be dominated by different termsat different distances. This can give rise to potentially observable cosmological signatures.In many cases the dominant states in the expansion do not correspond to small fluctuationsaround a minimum of the potential and are therefore not visible in perturbation theory. Wediscuss the physical interpretation these states, which can be present even when the potentialhas only one minimum. a r X i v : . [ g r- q c ] F e b ontents The study of a quantized scalar field in de Sitter space is a mature endeavour [1–4] possessingwell-known difficulties when the field is light [5–8]. Light spectator scalars in de Sitter spaceare not just of formal interest, but rather they can have a variety of cosmological implicationssuch as generation of dark matter [9–11] or triggering electroweak vacuum decay [12–14].The stochastic approach presented in Refs. [15, 16] is a powerful way of addressing thisproblem; it is analytically tractable yet it provides accurate results that are often superior tomore traditional resummation methods. The approach is based on the realization that theultraviolet part of the field may to a good approximation be treated as white noise allowingone to express all results via classical stochastics. For other techniques, see Refs. [17–29]The stochastic approach has become increasingly popular in recent years, likely due toits great efficacy, and in this vein we note the recent works [30–46]. Often the focus is onthe local probability distribution of the field or on local expectation values, even though thecorrelation of fluctutations over space is arguably the more relevant object physically. Thespatial correlators have been addressed for example in Refs. [9, 11, 35, 47–59]. Specifically, inRef. [59] it was shown how correlators at noncoincident points can be effectively calculatedwith numerical techniques in conjunction with the spectral expansion based on eigenfunctionsand -values already discussed in Ref. [15].The focus of Ref. [59] was on a single spectator scalar φ with quadratic and quarticterms in its potential while limiting the parameters to only include positive mass terms i.e.– 1 –otentials possessing a single minimum at the origin. In this work we extend this analysis toinclude potentials with two possibly non-degenerate minima. Namely, we focus on a potentialof the form V ( φ ) = µ φ + 12 m φ + λ φ , (1.1)with m ≤ µ ≥
0. Throughout we will make use of the parametrization ¯ α ≡ − m √ λH ; β ≡ µ λ / H ; ¯ m ≡ − m . (1.2)This potential has two minima if ¯ α > β/ / . In typical perturbative treatments, thefield is assumed to fluctuate near the minimum of the potential, but the stochastic spectralexpansion does not require that assumption. As we will show, in many cases the dominantcontributions to the correlators come from large-amplitude fluctuations that are not visiblein perturbation theory.This paper is organised as follows: In Section 2 we summarise the stochastic spectralexpansion and the numerical technique we used to find the terms in the expansion. In Sec-tions 3 and 4, we apply these techniques to calculate the spectral expansion in the symmetricand asymmetric cases, respectively. In Section 5 we discuss the physical interpretation andimplications of our results, and in Section 6 we summarise our conclusions. In the stochastic formalism one may express any correlator involving noncoincident spacetimepoints as a spectral expansion [16, 59], by solving the eigenvalues and eigenfunctions, Λ n and ψ n , from the eigenvalue equation D φ ψ n = − π Λ n H ψ n , (2.1)with D φ = 12 ∂ ∂φ − W ( φ ) ; v ( φ ) = 4 π H V ( φ ) ; W ( φ ) = v (cid:48) ( φ ) − v (cid:48)(cid:48) ( φ ) . (2.2)Since the input in the eigenvalue equation (2.1) is W ( φ ) (and not V ( φ )) it is the W ( φ ) thatwill turn out to be the fundamental quantity providing a qualitative understanding of thebehaviour of the eigenfunctions and -values. The behavior of v ( φ ) and W ( φ ) for the potential(1.1) is illustrated in Fig. 1. In most cases solving equation (2.1) needs numerical methods. If G f ( t , t ; x , x ) is the general correlator for some function of the field f ( φ ), the autocor-relation function G f ( t ; 0) = (cid:104) f ( φ (0)) f ( φ ( t )) (cid:105) , (2.3)can via the stochastic formalism be expressed as a spectral expansion [16, 59] making use ofthe eigenfunctions and -values from Eq. (2.1). Specifically, in terms of eigenvalues Λ n andspectral coefficients f n one may write G f ( t ; 0) = (cid:88) n f n e − Λ n t , (2.4) This is related to the definition in Ref. [59] as ¯ α ≡ − α . – 2 – − − − v ( φ ) ¯ α = 0 − − − ¯ α = 1 β = 0 β = 1 / β = 1 / − − − − − − − − ¯ α = 5 − φ − W ( φ ) ¯ α = 0 − φ ¯ α = 1 − − − φ − ¯ α = 5 Figure 1 . The v ( φ ) and W ( φ ) potentials as defined in (2.2) with the choice λ = 1 given in the units H = 1. where the spectral coefficients are specific to the form of f ( φ ) f n ≡ (cid:90) dφψ f ( φ ) ψ n . (2.5)By making use of the de Sitter invariance of the equilibrium quantum state, from (2.4) onemay infer the form of the equal time correlator between two spatially separated points G f (0; x ) = (cid:88) n f n ( | x | H ) n /H , (2.6)where x is physical. Following Ref. [59], we solve the eigenfunctions and eigenvalues of Eq. (2.1) numerically withthe ’overshoot/undershoot’ or otherwise known as ’wag the dog’ method. For a potential with Z symmetry this problem reduces to a systematic iteration of just one unknown variable,the eigenvalue Λ n . This is due to the general feature that for a symmetric potential thewave functions are either even or odd, which fixes one of the two initial conditions requiredfor a second order differential equation and the remaining one is irrelevant as the functionsmust be normalized. The essentials of the method are well-known and can be found for– 3 –xample in Section 2.3 of Ref. [60]. This special case is encoutered in the symmetric double-well potential, which is addressed in Sec. 3. A major shortcoming is that modifications arerequired when two eigenvalues are (almost) degenerate, which is encountered at the limit of alarge barrier, or equivalently, deep wells. However as we will show, this is precisely the limitfor which one may easily derive accurate analytic approximations and with a combination ofanalytics an numerics the entire eigenvalue spectrum may be covered.When there is no Z symmetry, for example as in Eq. (1.1) with β (cid:54) = 0 that is discussedin Sec. 4, the ’overshoot/undershoot’ needs to be performed with respect to two unknowns:in addition to the eigenvalue one must iterate over the value or derivative of the function ata point giving rise to a two dimensional iteration problem. This however amounts to only asmall increase in complexity of the algorithm or the expensiveness of the computation.To facilitate a numerical analysis of the eigenfunctions and -values very similarly toRef. [59] for the potential (1.1) it is convenient to introduce the following dimensionlessquantities z ≡ λ / Ω H φ, Ω ≡ (cid:18) mHλ / + µ λ / H (cid:19) ≡ (cid:16) √ ¯ α + β (cid:17) , (2.7)˜Λ n ≡ Λ n λ / H + ¯ m /H + µ /H ≡ Λ n λ / H (1 + ¯ α + β ) , (2.8)so that the eigenvalue equation (2.1) may be written in terms of dimensionless numbers as (cid:26) ∂ ∂z − π ¯ α + π β + 32 π ¯ αβz + 4 π − π ¯ α Ω z − π βz + 32 π ¯ αz − π z + 8 π (cid:0) α + β (cid:1) ˜Λ n Ω (cid:27) ψ n = 0 , (2.9)with the ¯ α and β given in (1.2). Furthermore and precisely as in Ref. [59] we also introducescaled eigenfunctions ψ n ≡ (cid:114) λ / Ω H ˜ ψ n ⇒ (cid:90) ∞−∞ dφ | ψ n | = (cid:90) ∞−∞ dz | ˜ ψ n | = 1 . (2.10)There are two reasons why the redefinitions we introduced are very useful for numericalwork. First, as one can see from Eq. (2.9), unlike the potential that is a function of µ , m and λ there are only two unknown parameters, ¯ α and β . Second, there are no terms in theequation that would grow without bound at any of the limiting cases ¯ α →
0, ¯ α → ∞ , β → β → ∞ , so broadly speaking all numerical factors are of the same order throughout theparameter space of interest. Before addressing the general situation, let us first focus on the important special case of asymmetric double-well potential with two degenerate minima i.e. with β = 0. The first fournon-trivial eigenvalues are plotted in Fig. 2. For ¯ α (cid:38) . and Λ becomedegenerate, complicating the numerical analysis, but by this point analytic approximationsderived in Section 3.2 (depicted with red dotted lines) are already very accurate.– 4 – ˜ φ ) ( ˜ φ ) ( ˜ φ ) ( ˜ φ ) n = 1 n = 2 n = 3 n = 4 − − ¯ α . . . . . ˜Λ n − − ¯ α − − | ( ˜ φ j ) n | Figure 2 . The lowest eigenvalues (left) and spectral coefficients for f ( φ ) = φ and f ( φ ) = φ (right)in the symmetric case, β = 0. The red dotted lines show the analytic approximations (3.9), (3.10),(3.11), (3.13) and (3.14). The eigenfunctions ˜ ψ n for n ≤ α (cid:38) W ( φ ), which can beseen in Fig. 1. Importantly, there are solutions that do not vanish close to the origin, evenwhen ¯ α (cid:29) ψ at ¯ α = 2 .
5, which is localised around the top of the potentialbarrier. As discussed in Section 5, these solutions can be interpreted as a contribution fromtransitions between the two minima, during which the field can spend a significant amount oftime near the top of the barrier. The existence of such solutions was apparently not noticedin Ref. [16].
As can be seen from Fig. 1, in the limit a large ¯ α the potential W ( φ ) in the eigenvalueequation (2.1) will possess three minima separated by large barriers, which can be shown tooccur at z = 0 ; z ± = ± √ ¯ α + 1 √ (cid:32)(cid:114) ¯ α + 274 π + 2 ¯ α (cid:33) / . (3.1)Hence, at the limit ¯ α → ∞ , the system separates into three quadratic pieces, which can beobtained from (2.9) by expanding around a large ¯ α with β = 0.For completeness we first write the eigenvalue equation for V ( φ ) = M φ (cid:26) ∂ ∂x − (cid:18) π (cid:19) x + 4 π π ˜Λ n (cid:27) ψ n = 0 ; x = MH φ ; ˜Λ n = Λ n M /H , (3.2) See Ref. [59] for more discussion and plots. – 5 – − − . . . . . ˜ ψ ¯ α = 0 − − − . . . . . ¯ α = 1 . − − − − . . . . . ¯ α = 2 . − − − . − . . . . . ˜ ψ ¯ α = 0 − − − . − . . . . . ¯ α = 1 . − − − − . − . . . . . ¯ α = 2 . − − − . − . . . . . . ˜ ψ ¯ α = 0 − − − . − . . . . . . ¯ α = 1 . − − − − . − . . . . . . ¯ α = 2 . − − − . − . . . . . ˜ ψ ¯ α = 0 − − − . − . . . . . ¯ α = 1 . − − − − . − . . . . . ¯ α = 2 . − − z − . − . . . . . ˜ ψ ¯ α = 0 − − z − . − . . . . . ¯ α = 1 . − − − z − . − . . . . . ¯ α = 2 . Figure 3 . The dimensionless eigenfunctions ˜ ψ n from Eq. (2.10) for the symmetric potential (1.1)with β = 0 as a function of ¯ α ≡ ¯ m / ( H √ λ ) The blue dashed lines indicate the locations of theminima of the potential V ( φ ). with the eigenfunctions and -values ψ n = √ MH √ n n ! (cid:18) π (cid:19) / e − π x H n (cid:18) πx √ (cid:19) ; Λ n = n M H . (3.3)Close to the origin at the limit of large barriers i.e. taking ¯ α → ∞ we then get theapproximate eigenvalue equation for the potential (1.1) (cid:26) ∂ ∂z − (cid:18) π (cid:19) z + 4 π π (cid:18) ˜Λ n − (cid:19)(cid:27) ψ n = 0 ; z = ¯ mH φ ; ˜Λ n = Λ n ¯ m /H , (3.4)– 6 –here the eigenfunctions and -values can be read off from the quadratic results (3.3) ψ n = √ ¯ mH √ n n ! (cid:18) π (cid:19) / e − π z H n (cid:18) πz √ (cid:19) ; Λ n = n + 13 ¯ m H . (3.5)At z ± for large barriers one gets the approximate equation (cid:26) ∂ ∂y − (cid:18) π (cid:19) y + 4 π π (cid:18) ˜Λ n (cid:19)(cid:27) ψ ± n = 0 ; y = √ (cid:0) z − z ± (cid:1) ; ˜Λ n = Λ n ¯ m /H , (3.6)with the eigenfunctions and -values ψ ± n = √ ¯ mH √ n n ! (cid:18) π (cid:19) / e − π y H n (cid:18) πy √ (cid:19) ; Λ n = 2 n m H . (3.7)By making use of the approximate solutions close to the origin and/or z ± , (3.5) and(3.7) respectively, and using Fig. 3 as a guide is it possible to understand qualitatively thelarge ¯ α behaviour and often write analytic approximations for the eigenfunctions and -values.Suppose an analytic function Φ n ( φ ) that approximates the full solution. Then, bycalculating the expectation value of the eigenvalue equation (2.1) one gets an approximationfor the eigenvalue as (cid:90) dφ Φ n D φ Φ n ≈ − π Λ n H . (3.8)For example, from Fig. 3 we see that for large ¯ α the n = 1 eigenfunction approachesan antisymmetric combination of two quadratic n = 0 solutions centered at z ± (3.7). Theeigenvalues of the quadratic n = 0 solutions are zero at z ± as given by (3.7), which impliesthat the first exited state has a neglibigle eigenvalue at this limit. As derived in [16], for anylarge but finite ¯ α the n = 1 eigenvalue is exponentially small.Similarly, we see that the eigenfunction for n = 2 approaches the quadratic n = 0solution at the origin, which as (3.5) shows does not result in a vanishing eigenvalue, even at¯ α (cid:29)
1. The analytic estimate for the eigenvalue is obtained by (cid:90) dφ ψ D φ ψ ≈ − π Λ H ⇒ ˜Λ ≈ π (cid:0) π ¯ α − (cid:1) ¯ α + 135768 π ¯ α ( ¯ α + 1) = 13 (cid:18) − α + 1 − π ¯ α (cid:19) + O ( ¯ α − ) . (3.9)The n = 3 case can be seen from Fig. 3 to approach a linear combination of threequadratic n = 1 solutions centered at the origin and at z ± . From (3.5) and (3.7) we see thatthe solutions are degenerate so we can derive an analytic approximation for the eigenvalueby simply using only the solution at z giving (cid:90) dφ ψ D φ ψ ≈ − π Λ H ⇒ ˜Λ ≈ π ¯ α − π ¯ α + 945768 π ¯ α ( ¯ α + 1) = 23 (cid:18) − α + 1 − π ¯ α (cid:19) + O ( ¯ α − ) . (3.10)– 7 –inally, the n = 4 eigenfunction clearly approaches a symmetric combination of twoquadratic n = 1 solutions centered at z ± , with then the eigenvalue at the large ¯ α limitapproximated by (cid:90) dφ ψ ± D φ ψ ± ≈ − π Λ H ⇒ ˜Λ ≈ (cid:18) − α + 1 + π ¯ α (cid:19) + O ( ¯ α − ) . (3.11)In the above for n = 4 we have only included the leading terms as the full result is quitecomplicated.The analytic approximations of this section are depicted by the red dashed curves inthe left panel in Fig. 2. As one may see, they are in very good agreement with the full resultsfor ¯ α (cid:38) In addition to the eigenvalues, the spectral coefficients are the other ingredient needed forcalculating correlators as given by Eq. (2.4) and (2.6). As an illustration in the following weanalyse the leading and next-to-leading spectral coefficients for f ( φ ) = φ and f ( φ ) = φ . Auseful dimensionless definition comes via (see Eq. (2.10))( ˜ φ j ) n ≡ (cid:18) λ / Ω H (cid:19) j ( φ j ) n = (cid:18) λ / Ω H (cid:19) j (cid:90) ∞−∞ dφψ φ j ψ n = (cid:90) ∞−∞ dz ˜ ψ z j ˜ ψ n . (3.12)Much like for the eigenvalues, the spectral coefficients can in some circumstances be approx-imated with analytic results at the large barrier limit, which may be deduced from section3.2 and in particular Fig. 3.Let us focus on f ( φ ) = φ first. Because it is an odd function, only odd n contribute.Because the ground state and the first exited state approach symmetric and antisymmetriccombinations of the ground state of a harmonic oscillator located at z ± , we may approximatethe ( φ ) coefficient as | ( φ ) | ≈ | (cid:90) ∞−∞ dφ √ (cid:0) ψ +0 + ψ − (cid:1) φ √ (cid:0) ψ − − ψ +0 (cid:1) | ≈ | (cid:90) dφφ (cid:0) ψ ± (cid:1) |⇒ | ( ˜ φ ) | ≈ | (cid:90) dzz (cid:0) ˜ ψ ± (cid:1) | = (cid:0) √ ¯ α + 1 (cid:1) (cid:112) √ π ¯ α + 27 + 4 π ¯ α √ π . (3.13)The third excited state is for large ¯ α approximately a linear combination of ψ and − ψ +1 − ψ − . For this case in our approximative prescription there is an ambiguity as thereis no a priori way to determine the relative size between ψ and − ψ +1 − ψ − .The analytic approximations (3.13) and (3.14), as well as the numerical results for theleading and next-to-leading spectral coefficients are shown in Fig. 2 . The | ( ˜ φ ) | term, shownby the dashed green line, is cut short by our inability to extend the numerical method tocases with almost degenerate eigenvalues, which occurs for n = 3. However, it is clearlysubleading to | ( ˜ φ ) | denoted with green that does not suffer from this issue.Correspondingly, because f ( φ ) = φ is even, only even values of n contribute to itscorrelator. Since there is virtually no overlap with the n = 0 and n = 2 solutions at the largebarrier limit, | ( ˜ φ ) | is expected to vanish up to exponentially small terms. In contrast, ( φ ) – 8 – ˜ φ ) ( ˜ φ ) ( ˜ φ ) ( ˜ φ ) n = 1 n = 2 n = 3 n = 4 − − ¯ α . . . . . . ˜Λ n β = 0 . − − ¯ α − − | ( ˜ φ j ) n | β = 0 . Figure 4 . The eigenvalues ˜Λ n (left) and the spectral coefficients for f ( φ ) = φ (right) in theasymmetric case with β = 0 .
5, as functions of ¯ α . has the approximation | ( φ ) | ≈ | (cid:90) ∞−∞ dφ √ (cid:0) ψ +0 + ψ − (cid:1) φ √ (cid:0) ψ +1 − ψ − (cid:1) | ≈ | (cid:90) dφφ ψ ± ψ ± |⇒ | ( ˜ φ ) | ≈ | (cid:90) dzz ˜ ψ ± ˜ ψ ± | = (cid:0) √ ¯ α + 1 (cid:1) (cid:113) π + √ π ¯ α +27¯ α π / . (3.14)This, together with the numerical results for | ( ˜ φ ) | and | ( ˜ φ ) | are shown in Fig. 2 . The factthat | ( φ ) | can dominate over | ( φ ) | as seen in Fig. 2 has important physical consequences,which are discussed in Section 5. Let us now move to the more general asymmetric case, with β >
0. Now, the potential V ( φ )has a single minimum if ¯ α < β/ / , and two non-degenerate minima if ¯ α > β/ / .On the hand, the function W ( φ ), which appears in the eigenvalue equation (2.1), seems toalways have several minima. Depending on its specific form, some of the lowest eigenfunctionsmay be localised at the other minima of W ( φ ), rather than the vacuum state located atthe minimum of the potential V ( φ ). These contributions would not be visible in typicalperturbation theory calculations. They can also give rise to non-trivial hierarchies betweenthe spectral coefficients (2.5).As an illustrative example, Fig. 4 shows the lowest eigenvalues for β = 0 . f ( φ ) = φ . At ¯ α = 0, the lowest eigenvalue Λ islocalised around the vacuum state and therefore it corresponds to perturbative fluctuations.– 9 – − − − ˜ ψ n n = ˜Λ = 0 − − − − n = 1 ; ˜Λ = 0 . − − − − n = 2 ; ˜Λ = 0 . − − − − ˜ ψ n n = 3 ; ˜Λ = 0 . − − − − n = 4 ; ˜Λ = 0 . ¯ α = 1 β = 0 . Figure 5 . The n ≤ α = 1 and β = 0 .
5. The blue dashed line indicates the location of the minimum of the potential.
However, when ¯ α becomes larger, it is overtaken by other eigenvalues which are localisedaround the other minima of W ( φ ).This can be seen in Fig. 5, which shows the five lowest eigenfunctions for ¯ α = 1 and β = 0 .
5. Even though the potential V ( φ ) actually has only one minimum for these parameters(see Fig. 1), the lowest excited state localised at the minimum of the potential is n = 4. Theasymptotic form of the correlator is therefore determined by the shape of the potential awayfrom its minimum. However, because such states have a very small overlap with the groundstate ψ , their spectral coefficients are very small. This is discussed more in Section 5. Theeigenfunctions and -values up to n = 4 for some representative choices for ¯ α and β are shownin Figs. 8–12 in Appendix A. In cosmology, the observable length scales correspond to comoving distances that were manyorders of magnitude longer than the Hubble length during inflation. We are therefore ofteninterested in the correlator at distances that are very long but still finite.The asymptotic long-distance behaviour of the correlator G f (0 , r ) is given by the firstterm in the spectral expansion (2.6) with a non-zero spectral coefficient f n . The behaviouris simplest if the lowest state, n = 1, has the largest spectral coefficient, because then thecorrelator is well approximated by a single power-law at all length scales, G f (0 , r ) ≈ f ( rH ) /H . (5.1)However, this is not always the case, and more generally, the correlator can be dominated bya higher term n = d in the expansion at the distances of interest, G f (0 , r ) ≈ f d ( rH ) d /H . (5.2)– 10 –n particular, the short-distance behaviour of the correlator can be very different fromits asymptotic long-distance form. To characterise that, following Ref. [16] we define acorrelation radius to be the distance where the correlator has fallen to half of its value at r = 1 /H , G f (0 , R f ) = 12 G f (0 , /H ) . (5.3)When a single coefficient n = d dominates the correlator, this simply gives R f ≈ H − H d . (5.4)The non-trivial hierarchies between different terms in the spectral expansion can be importantfor cosmological observations. If there is a change in the behaviour of the correlator withinthe observable scales, it can potentially be detected providing useful information about thefields responsible for it. In the symmetric case ( β = 0) discussed in Section 3, there are two examples of this non-trivial behaviour. The first is that, because the lowest eigenstate n = 1 has odd parity,the corresponding spectral coefficient vanishes for all even functions. The asymptotic long-distance behaviour of any even correlator, such as those of φ or the energy density, istherefore given by the second-lowest eigenvalue Λ . When ¯ α (cid:38)
1, the difference betweenthem can be large, as we can see from the left panel in Fig. 2. Because Λ is small, the fielditself is correlated over massively superhorizon scales, but its energy density is not, becauseits correlations are determined by Λ . The physical reason for this behaviour is that onsuperhorizon scales, the system consists of domains of the two vacua, which contribute tothe field correlator, and the correlation radius R φ gives the typical size of these domains.However, because both vacua have the same energy density, these domains do not give anycontribution to correlators of even quantities such as the energy density.The second example is that, as we can see from the right panel of Fig. 2, the spectralcoefficient of φ for the second-lowest state n = 2 falls rapidly when ¯ α (cid:38)
1. This means thatalthough the asymptotic long-distance behaviour of any even correlator is indeed given by Λ ,it only starts to dominate at very long superhorizon distances, and at shorter distances thedominant contribution is given by Λ . To understand why this happens, it is useful to lookat the corresponding eigenfunctions in Fig. 3. The higher state ψ is localised at the minimaof the potential, and therefore it corresponds to small-amplitude perturbative fluctuationsaround either vacuum state. The lower state ψ , on the other hand, is localised on top ofthe barrier, φ = 0, which shows that this contribution comes from the boundaries betweenthe domains. The eigenvalue Λ characterises the thickness of these domain walls, and thespectral coefficient is small because their volume is small compared with the volume of thedomains.The different behaviour of odd and even correlators is illustrated by Fig. 6, which showsthe correlation radii of φ and δφ . As Λ becomes small, the field correlation radius grows tomassively superhorizon scales. In contrast, when ¯ α (cid:38)
1, the correlation radius of δφ startsto decrease, first because Λ grows, and then because Λ starts to dominate it.If any variable has significant correlations at distances much longer than the currentHubble length 1 /H , they would appear as a homogeneous background value. In the case ofa symmetric potential, this happens when the domain size is larger than 1 /H , in which case– 11 – = 0 . λ = 0 . λ = 0 . − − ¯ α f ( φ ) = δφ − − ¯ α R f H f ( φ ) = φ Figure 6 . The correlation radii for f ( φ ) = φ and f ( φ ) = δφ ≡ φ − (cid:104) φ (cid:105) for three values of λ . the current observable Universe would most likely be inside single domain. In that case theobserved field correlator would be determined by the second-lowest odd eigenvalue Λ . In the asymmetric case discussed in Section 4, we can see further examples of these non-trivialhierarchies. When ¯ α is small, the lowest state n = 1 corresponds to perturbative fluctuationsaround the true vacuum. However, as ¯ α increases, other states overtake it one by one, as wecan see from Fig. 4.Fig. 5 shows the eigenfunctions for ¯ α = 1 and β = 0 .
5, and illustrates that for theseparameters the lowest excited state that has significant overlap with the vacuum is n = 4.Because of this, this state continues to have the highest spectral coefficient, and therefore itdominates the correlator at short distances. On the other hand, the asymptotic long-distancebehaviour is given by the lowest state Λ . Because the corresponding spectral coefficient isvery small, it only starts to dominate the correlator at extremely long super-horizon distances.When the potential V ( φ ) has two non-degenerate minima separated by a high barrier,the lowest state n = 1 is localised in the false vacuum state, and therefore we can interpretit as the contribution from the domains of false vacuum which are occasionally formed. Theeigenvalue Λ represents the size of these domains, and the spectral coefficient is suppressedbecause of their rarity.However, in the example shown in Fig. 5 this behaviour occurs even though the potential V ( φ ) has only one minimum and there is therefore no false vacuum state (see Fig. 1). Inthat case the lowest state n = 1 corresponds to excursions of the field into high field valuesfar away from the minimum.As in the asymmetric case, it is possible that Λ is so small that the false vacuumdomains are larger than 1 /H . Then, because the true and false vacuum have differentphysical properties, the observables we would measure would depend on the vacuum we are– 12 – = 0 . λ = 0 . λ = 0 . λ = 0 . − − ¯ α − − − − k ∗ H n δφ − ≈ Hn δφ − ≈ H Figure 7 . The wavenumber k ∗ at which the P (2) δφ ( k ) term starts dominating the power spectrum(5.5) over P (4) δφ ( k ). Above the gray line the distances are sub-horizon and hence not amendable to astochastic treatment. in. This would not affect the eigenvalues in the spectral expansion (2.6), but the spectralcoefficients f n would have to be computed using the false vacuum ground state ψ ratherthan the true ground state ψ . For cosmological observations, the power spectrum is often a more relevant quantity than thecoordinate-space correlator. Therefore it is useful to calculate the power spectrum by takingthe Fourier transform of (2.6) P f ( k ) = (cid:88) n π f n Γ (cid:18) − n H (cid:19) sin (cid:18) Λ n πH (cid:19)(cid:18) kH (cid:19) n /H ≡ (cid:88) n =1 P ( n ) f ( k ) , (5.5)where k is the physical momentum.Assuming that a single n = d term dominates the expansion, the spectral index n f canthen be written in a simple form n f − ≡ ln P f ( k )ln k ≈ ln P ( d ) f ( k )ln k = 2Λ d H . (5.6)When there are non-trivial hierarchies between the spectral coefficients, the spectralindex n f can be different on different scales. As an example, consider the power spectrumof δφ ≡ φ − (cid:104) φ (cid:105) in the symmetric case. At asymptotically small wave number k , thespectral index is given by the lowest state n = 2, but because its spectral coefficient is verysmall, the higher state n = 4 dominates at higher k . Comparing these two terms, we can– 13 –traightforwardly solve for the wavenumber k ∗ at which they cross as P (2) δφ ( k ∗ ) P (4) δφ ( k ∗ ) = 1 ⇔ k ∗ H = (cid:40) (cid:2) ( φ ) (cid:3) Γ (cid:0) − Λ H (cid:1) sin (cid:0) Λ πH (cid:1) [( φ ) ] Γ (cid:0) − Λ H (cid:1) sin (cid:0) Λ πH (cid:1) (cid:41) H − Λ2) , (5.7)which is plotted in Fig. 7. When ¯ α (cid:46) k ∗ /H (cid:38)
1, and therefore the first term n = 2dominates on all scales and the spectral index is therefore constant to a good approximation.However, when ¯ α (cid:38)
1, we can see that k ∗ /H rapidly becomes very small, which means thecrossover happens at scales that were massively superhorizon during inflation. What thisshows is that the scale at which the spectral index changes can for some parameter valuesoccur at scales that are visible in the cosmic microwave background or other cosmological ob-servations, possibly providing an important observational signature of early Universe modelsinvolving decoupled spectator scalars. In this work we have studied spectator scalar fields in de Sitter space with a potential of thedouble-well form (1.1) by means of the stochastic spectral expansion [16]. The terms in theexpansion are determined by eigenvalues and eigenfunctions which we solve numerically. Thiswork is a continuation of Ref. [59] where only potentials with manifest Z symmetry and asingle minimum were considered. Our calculations show that also asymmetric potentials withmultiple minima can be efficiently studied with simple numerical methods to high precision,implying the technique to be rather powerful and suitable for a large class of potentials. Inthis vein we note two interesting possibilities that are yet unexplored: models with morethan one scalar and/or potentials with periodic boundary conditions such as for the axion.The double-well potential has unsurprisingly a much richer structure in terms of eigen-functions and -values than the quartic or quadratic cases. In particular, the spectrum hasstates that are not localised near the minimum of the potential and which are thereforenot visible in perturbation theory. In most cases these non-perturbative states dominatethe correlation function especially at long distances. The spectral coefficients of the termscan also be very different, which can lead to non-trivial behaviour such as a change of thespectral index at very long distances. If this happens at cosmologically relevant scales, itcan be observable. Interestingly, this behaviour persists even when the potential has onlyone minimum. In the symmetric case, the behaviour of odd quantities (such as the field) isalso very different from the behaviour of even quantities (such as energy density). Together,these effects demonstrate that the naive intuition based perturbation theory is, in general,not applicable.Based on our analysis we conclude that spectator fields with a double-well potentialhave many possibly observable and currently unexplored cosmological consequences. Acknowledgments
We thank Jacopo Fumagalli, Gabriel Moreau, Julien Serreau and S´ebastien Renaux-Petelfor illuminating discussions. AR is supported by the U.K. Science and Technology FacilitiesCouncil grant ST/P000762/1 and an IPPP Associateship. This project has received fundingfrom the European Union’s Horizon 2020 research and innovation programme under theMarie Sk(cid:32)lodowska-Curie grant agreement No. 786564.– 14 – − − . . . . . ˜ ψ ¯ α = 0 β = 0 − − − . . . . . ¯ α = 0 . β = 0 − − − − . . . . . ¯ α = 2 β = 0 − − − . . . . . . . ˜ ψ ¯ α = 0 β = 0 . − − − . . . . . . . ¯ α = 0 . β = 0 . − − − − . . . . . . . ¯ α = 2 β = 0 . − − z − . . . . . . . ˜ ψ ¯ α = 0 β = 0 . − − z − . . . . . . . ¯ α = 0 . β = 0 . − − − z − . . . . . . . ¯ α = 2 β = 0 . Figure 8 . The dimensionless n = 0 eigenfunctions from Eq. (2.9). A Eigenfunctions and -values
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