A Density Spike on Astrophysical Scales from an N-Field Waterfall Transition
Illan F. Halpern, Mark P. Hertzberg, Matthew A. Joss, Evangelos I. Sfakianakis
aa r X i v : . [ a s t r o - ph . C O ] J u l MIT-CTP 4499
A Density Spike on Astrophysical Scales from an N -Field Waterfall Transition Illan F. Halpern , , ∗ , Mark P. Hertzberg , † , Matthew A. Joss , ‡ , Evangelos I. Sfakianakis , , § Center for Theoretical Physics and Dept. of Physics,Massachusetts Institute of Technology, Cambridge, MA 02139, USA Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Dept. of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (Dated: September 30, 2018)Hybrid inflation models are especially interesting as they lead to a spike in the density powerspectrum on small scales, compared to the CMB, while also satisfying current bounds on tensormodes. Here we study hybrid inflation with N waterfall fields sharing a global SO ( N ) symmetry.The inclusion of many waterfall fields has the obvious advantage of avoiding topologically stabledefects for N >
3. We find that it also has another advantage: it is easier to engineer models thatcan simultaneously (i) be compatible with constraints on the primordial spectral index, which tendsto otherwise disfavor hybrid models, and (ii) produce a spike on astrophysically large length scales.The latter may have significant consequences, possibly seeding the formation of astrophysically largeblack holes. We calculate correlation functions of the time-delay, a measure of density perturbations,produced by the waterfall fields, as a convergent power series in both 1 / N and the field’s correlationfunction ∆( x ). We show that for large N , the two-point function is h δt ( x ) δt ( ) i ∝ ∆ ( | x | ) / N andthe three-point function is h δt ( x ) δt ( y ) δt ( ) i ∝ ∆( | x − y | )∆( | x | )∆( | y | ) / N . In accordance with thecentral limit theorem, the density perturbations on the scale of the spike are Gaussian for large N and non-Gaussian for small N . CONTENTS
I. Introduction 1II. N Field Model 2A. Approximations 3B. Mode Functions 4III. The Time-Delay 4IV. Correlation Functions 5A. Two-Point Function 6B. Three-Point Function 7C. Momentum Space 8V. Constraints on Hybrid Models 9A. Topological Defects 9B. Inflationary Perturbations 9C. Implications for Scale of Spike 10D. Eternal Inflation 11VI. Discussion and Conclusions 11A. Series Expansion to Higher Orders 12B. Two-Point Function for Even Number of Fields 12C. Alternative Derivation of Time-Delay Spectra 13References 13 ∗ Electronic address: [email protected] † Electronic address: [email protected]
I. INTRODUCTION
Inflation, a phase of accelerated expansion in the veryearly universe thought to be driven by one or severalscalar fields, is our paradigm of early universe cosmology[1–4]. It naturally explains the large scale homogeneity,isotropy, and flatness of the universe. Moreover, the ba-sic predictions of even the simplest single field slow rollmodels, giving approximate scale invariance and smallnon-Gaussianity in the ∼ − level departures from ho-mogeneity and isotropy, are in excellent agreement withrecent CMB data [5–7] and large scale structure.While the basic paradigm of inflation is in excellentshape, no single model stands clearly preferred. In-stead the literature abounds with various models mo-tivated by different considerations, such as string mod-uli, supergravity, branes, ghosts, Standard Model, etc[8–19]. While the incoming data is at such an impres-sive level that it can discriminate between various mod-els and rule out many, such as models that overpredictnon-Gaussianity, it is not clear if the data will ever revealone model alone. An important way to make progress isto disfavor models based on theoretical grounds (such asissues of unitarity violation, acausality, etc) and to finda model that is able to account for phenomena in theuniverse lacking an alternate explanation. It is conceiv-able that some version of the so-called “hybrid inflation”model may account for astrophysical phenomena, for rea-sons we shall come to. ‡ Electronic address: [email protected] § Electronic address: [email protected]
The hybrid inflation model, originally proposed byLinde [20], requires at least two fields. One of the fieldsis light and another of the fields is heavy (in Hubbleunits). The light field, called the “timer”, is at earlytimes slowly rolling down a potential hill and generatesthe almost scale invariant spectrum of fluctuations ob-served in the CMB and in large scale structure. Theheavy field, called the “waterfall” field, has an effectivemass that is time-dependent and controlled by the valueof the timer field. The waterfall field is originally trappedat a minimum of its potential, but as its effective mass-squared becomes negative, a tachyonic instability follows,leading to the end of inflation; an illustration is given inFigure 1. The name “hybrid inflation” comes from thefact that this model is a sort of hybrid between a chaoticinflation model and a symmetry breaking inflation model.As originally discussed in Refs. [21, 22], one of the mostfascinating features of hybrid models is that the tachyonicbehavior of the waterfall field leads to a sharp “spike” inthe density power spectrum. This could seed primor-dial black holes [23–28]. For generic parameters, thelength scale associated with this spike is typically verysmall. However, if one could find a parameter regimewhere the waterfall phase were to be prolonged, lastingfor many e-foldings, say N w ∼ −
40, then this wouldlead to a spike in the density perturbations on astro-physically large scales (but smaller than CMB scales).This may help to account for phenomena such as super-massive black holes or dark matter, etc. Of course thedetails of all this requires a very careful examination ofthe spectrum of density perturbations, including obser-vational constraints.Ordinarily the spectrum of density perturbations in agiven model of inflation is obtained by decomposing theinflaton field into a homogeneous part plus a small in-homogeneous perturbation. However, for the waterfallfields of hybrid inflation, this approach fails since clas-sically the waterfall field would stay forever at the topof a ridge in its potential. It is the quantum perturba-tions themselves that lead to a non-trivial evolution ofthe waterfall field, and therefore the quantum perturba-tions cannot be treated as small. Several approximationshave been used to deal with this problem [21, 22, 29–40].Here we follow the approach presented by one of us re-cently in Ref. [41], where a free field time-delay methodwas used, providing accurate numerical results.In this paper, we generalize the method of Ref. [41] toa model with N waterfall fields sharing a global SO ( N )symmetry. A model of many fields may be natural invarious microscopic constructions, such as grand unifiedmodels, string models, etc. But apart from generalizingRef. [41] to N fields, we also go much further in our anal-ysis: we derive explicit analytical results for several corre-lation functions of the so-called time-delay. We formulatea convergent series expansion in powers of 1 / N and thefield’s correlation function ∆( x ). We find all terms in theseries to obtain the two-point correlation function of thetime-delay for any N . We also obtain the leading order behavior at large N for the three-point function time-delay, which provides a measure of non-Gaussianity. Wefind that the non-Gaussianity is appreciable for small N and suppressed for large N .We also analyze in detail constraints on hybrid infla-tion models. We comment on how multiple fields avoidstopological defects, which is a serious problem for low N models. However, the most severe constraint on hy-brid models comes from the requirement to obtain theobserved spectral index n s . We show that at large N ,it is easier to engineer models that can fit the observed n s , while also allowing for a prolonged waterfall phase.This means that large N models provide the most plau-sible way for the spike to appear on astrophysically largescales and be compatible with other constraints.With regards to tensor modes, the confirmation of therecent detection and amplitude of primordial gravita-tional waves [42] depends largely on understanding dustforegrounds [43]. The final answer to the existence ofprimordial tensor modes will have to be given by futureexperiments, since a recent joint analysis of current data[44] does not provide a clear detection. So we will onlybriefly discuss possibilities of obtaining observable tensormodes from hybrid inflation in Section V.The organization of our paper is as follows: In Sec-tion II we present our hybrid inflation model and discussour approximations. In Section III we present the time-delay formalism, adapting the method of Ref. [41] to N fields. In section IV we derive a series expansion for thetwo-point function, we derive the leading order behav-ior of the three-point function, and we derive results in k -space. In Section V we present constraints on hybridmodels, emphasizing the role that N plays. In SectionVI we discuss and conclude. Finally, in the Appendiceswe present further analytical results. II. N FIELD MODEL
The model consists of two types of fields: The timerfield ψ that drives the first slow-roll inflation phase, andthe waterfall field φ that becomes tachyonic during thesecond phase causing inflation to end. In many hybridmodels, φ is comprised of two components, a complexfield, but here we allow for N real components φ i . Weassume the components share a global SO ( N ) symmetry,and so it is convenient to organize them into a vector ~φ = { φ , φ , . . . , φ N } . (1)For the special case N = 2, this can be organized into acomplex field by writing φ complex = ( φ + i φ ) / √ ψ, ~φ minimally coupled togravity as follows (signature + − −− ) S = Z d x √− g " πG R + 12 g µν ∂ µ ψ∂ ν ψ + 12 g µν ∂ µ ~φ · ∂ ν ~φ − V ( ψ, φ ) . (2)The potential V is given by a sum of terms: V pro-viding false vacuum energy, V ψ ( ψ ) governing the timerfield, V φ ( φ ) governing the waterfall field, and V int ( ψ, φ )governing their mutual interaction, i.e., V ( ψ, φ ) = V + V ψ ( ψ ) + V φ ( φ ) + V int ( ψ, φ ) . (3)During inflation we assume that the constant V domi-nates all other terms.The timer field potential V ψ and the waterfall field po-tential V φ can in general be complicated. In general,they are allowed to be non-polynomial functions as partof some low energy effective field theory, possibly fromsupergravity or other microscopic theories. For our pur-poses, it is enough to assume an extremum at ψ = φ = 0and expand the potentials around this extremum as fol-lows: V ψ ( ψ ) = 12 m ψ ψ + . . . (4) V φ ( φ ) = − m ~φ · ~φ + . . . (5)The timer field is assumed to be light m ψ < H and thewaterfall field is assumed to be heavy m > H , where H is the Hubble parameter. In the original hybrid model,this quadratic term for V ψ was assumed to be the entirepotential. This model leads to a spectrum with a spectralindex n s > m ψ , which has important consequences.We discuss these issues in detail in Section V.As indicated by the negative mass-squared, the water-fall field is tachyonic around φ = ψ = 0. This obviouslycannot be the entire potential because then the potentialwould be unbounded from below. Instead there mustbe higher order terms that stabilize the potential witha global minimum near V ≈ V int ( ψ, φ ) = 12 g ψ ~φ · ~φ. (6)This term allows the waterfall field to be stabilized at φ = 0 at early times during slow-roll inflation when ψ is displaced away from zero, and then to become tachy-onic once the timer field approaches the origin; this isillustrated in Figure 1. FIG. 1. An illustration of the evolution of the effective poten-tial for the waterfall field φ as the timer field ψ evolves from“high” values at early times, to ψ = ψ c , and finally to “low”values at late times. In the process, the effective mass-squaredof φ evolves from positive, to zero, to negative (tachyonic). A. Approximations
We assume that the constant V is dominant duringinflation, leading to an approximate de Sitter phase withconstant Hubble parameter H . By assuming a flat FRWbackground, the scale factor is approximated as a = exp( Ht ) . (7)At early times, ψ is displaced from its origin, so φ = 0.This means that we can approximate the ψ dynamics byignoring the back reaction from φ . The fluctuations in ψ establish nearly scale invariant fluctuations on largescales, which we shall return to in Section V. However,for the present purposes it is enough to treat ψ as a classi-cal, homogeneous field ψ ( t ). We make the approximationthat we can neglect the higher order terms in the poten-tial V ψ in the transition era, leading to the equation ofmotion ¨ ψ + 3 H ˙ ψ + m ψ ψ = 0 . (8)Solving this equation for ψ ( t ), we insert this into theequation for φ i . We allow spatial dependence in φ i , andignore, for simplicity, the higher order terms in V φ , lead-ing to the equation of motion¨ φ i + 3 H ˙ φ i − e − Ht ∇ φ i + m φ ( t ) φ i = 0 . (9)Here we have identified an effective mass-squared for thewaterfall field of m φ ( t ) ≡ − m − (cid:18) ψ ( t ) ψ c (cid:19) ! , (10)where the dimension 4 coupling g has been traded forthe parameter ψ c as g = m /ψ c . The quantity ψ c hasthe physical interpretation as the “critical” value of ψ such that the effective mass of the waterfall field passesthrough zero. So at early times for ψ > ψ c , then m φ > φ i is trapped at φ i = 0, while at late times for ψ <ψ c , then m φ < φ i is tachyonic and can grow inamplitude, depending on the mode of interest; Figure 1illustrates these features. B. Mode Functions
Since we ignore the back-reaction of φ onto ψ and sincewe treat ψ as homogeneous in the equation of motion for φ (eq. (9)), then by passing to k -space, all modes aredecoupled. Each waterfall field φ i can be quantized andexpanded in modes in momentum space as follows φ i ( ~x, t ) = Z d k (2 π ) h c k,i e i k · x u k ( t ) + c † k,i e − i k · x u ∗ k ( t ) i , (11)where c † k ( c k ) are the creation (annihilation) operators,acting on the φ i = 0 vacuum. By assuming an initialBunch-Davies vacuum for each φ i , the mode functions u k are the same for all components due to the SO ( N )symmetry. To be precise, we assume that at asymptot-ically early times the mode functions are the ordinaryMinkowski space mode functions, with the caveat thatwe need to insert factors of the scale factor a to convertfrom physical wavenumbers to comoving wavenumbers,i.e., u k ( t ) → e − i k t/a a √ k , at early times . (12)At late times the mode functions behave very differently.Since the field becomes tachyonic, the mode functionsgrow exponentially at late times. The transition dependson the wavenumber of interest. The full details of themode functions were explained very carefully in Ref. [41],and the interested reader is directed to that paper formore information.Since we are approximating φ i as a free field here, itsfluctuations are entirely Gaussian and characterized en-tirely by its equal time two-point correlation function h φ i ( x ) φ j ( y ) i . Passing to k -space, and using statisticalisotropy and homogeneity of the Bunch-Davies vacuum,the fluctuations are equally well characterized by the so-called power spectrum P φ ( k ), defined through h φ i ( k ) φ j ( k ) i = (2 π ) δ ( k + k ) δ ij P φ ( k ) . (13)This means the power spectrum is δ ij P φ ( k ) = Z d x e − i k · x h φ i ( x ) φ j ( ) i . (14)It is simple to show that the power spectrum is relatedto the mode functions by P φ ( k ) = | u k | . (15) FIG. 2. A plot of the (re-scaled) field’s power spectrum P ˜ φ as a function of wavenumber k (in units of H ) for differentmasses: blue is m = 2 and m ψ = 1 /
2, red is m = 4 and m ψ = 1 /
2, green is m = 2 and m ψ = 1 /
4, orange is m = 4and m ψ = 1 / In Figure 2 we plot a rescaled version of P φ , where wedivide out by the root-mean-square (rms) of φ defined as φ rms = h φ i (0) i . As will be mentioned in the next Sectionand is extensively discussed in [41], the ratio˜ φ ( t ) = φ ( t ) φ rms ( t ) (16)is time-independent for late times, hence so is P ˜ φ ( k ). III. THE TIME-DELAY
We would now like to relate the fluctuations in the wa-terfall field φ to a fluctuation in a physical observable,namely the density perturbation. An important step inthis direction is to compute the so-called “time-delay” δt ( x ); the time offset for the end of inflation for differentparts of the universe. This causes different regions of theuniverse to have inflated more than others, creating adifference in their densities (though we will not explicitlycompute δρ/ρ here). This basic formalism was first intro-duced by Hawking [45] and by Guth and Pi [46], and hasrecently been reviewed in Ref. [47], where the transitionfrom the time-delay formalism to the more frequentlyused curvature perturbation R is outlined. In the con-text of hybrid inflation, it was recently used by one ofus in Ref. [41]. It provides an intuitive and straightfor-ward way to calculate primordial perturbations and wenow use this to study perturbations established by the N waterfall fields.In its original formulation, the time-delay formalismstarts by considering a classical homogeneous trajectory φ = φ ( t ), and then considers a first order perturbationaround this. At first order, one is able to prove that thefluctuating inhomogeneous field φ ( x , t ) is related to theclassical field φ , up to an overall time offset δt ( x ), φ ( x , t ) = φ ( t − δt ( x )) . (17)In the present case of hybrid inflation, the waterfallfield is initially trapped at φ = 0 and then once it be-comes tachyonic, it eventually falls off the hill-top due toquantum fluctuations. This means that there is no clas-sical trajectory about which to expand. Nevertheless, wewill show that, to a good approximation, the field φ ( x , t )is well described by an equation of the form (17), for asuitably defined φ . The key is that all modes of interestgrow at the same rate at late times. Further informationof the time evolution of the mode functions can be foundin Ref. [41].To show this, we need to compute the evolution of thefield φ according to eq. (9). This requires knowing m φ ( t ),which in turn requires knowing ψ ( t ). By solving eq. (8)for the timer field and dispensing with transient behavior,we have ψ ( t ) = ψ c exp ( − p t ) , (18)where p = H − s − m ψ H , (19)(note p > t = 0 to bewhen ψ = ψ c and assume ψ > ψ c at early times.Substituting this solution into m φ ( t ), we can, in prin-ciple, solve eq. (9). In general, the solution is somewhatcomplicated with a non-trivial dependence on wavenum-ber. However, at late times the behavior simplifies. Ourmodes of interest are super-horizon at late times. Forthese modes, the gradient term is negligible and the equa-tion of motion reduces to¨ φ i + 3 H ˙ φ i + m φ ( t ) φ i = 0 . (20)So each mode evolves in the same way at late times.Treating m φ ( t ) as slowly varying (which is justified be-cause the timer field mass m ψ < H and so p is small), wecan solve for φ i at late times t in the adiabatic approxi-mation. We obtain φ i ( x , t ) = φ i ( x , t ) exp (cid:18)Z tt dt ′ λ ( t ′ ) (cid:19) , (21)where λ ( t ) = H −
32 + r
94 + m H (cid:0) − e − p t (cid:1)! . (22)Here t is some reference time. For t >
0, we have λ ( t ) >
0, so the modes grow exponentially in time. Later inSection V we explain that in fact λ is roughly constant in the latter stage of the waterfall phase, i.e., the exp( − p t )piece becomes small.We now discuss fluctuations in the time at which in-flation ends. For convenience, we define the referencetime t to be the time at which the rms value of the fieldreaches φ end ; the end of inflation N φ ( t ) = φ , (23)where we have included a factor of N to account for allfields, allowing φ rms to refer to fluctuations in a singlecomponent φ i , i.e., φ rms = h φ i (0) i . In terms of the powerspectrum, it is φ rms = Z d k (2 π ) P φ ( k ) . (24)If we were to include arbitrarily high k , this would di-verge quadratically, which is the usual Minkowski spacedivergence. However, our present analysis only applies tomodes that are in the growing regime. For these modes, P φ ( k ) falls off exponentially with k and there is no prob-lem. So in this integral, we only include modes thatare in the asymptotic regime, or, roughly speaking, onlysuper-horizon modes.Using eq. (21), we can express the field φ i ( x , t ) at time t = t + δt in terms of the field φ i ( x , t ) by φ i ( x , t ) = φ i ( x , t ) exp ( λ ( t ) δt ) . (25)If t is chosen to be the time t end ( x ) at which inflation endsat each point in space, then ~φ · ~φ (cid:0) x , t end ( x ) (cid:1) = φ = N φ ( t ), and the above equation becomes N φ ( t ) = ~φ ( x , t ) · ~φ ( x , t ) exp (2 λ ( t ) δt ) , (26)which can be solved for the time-delay field δt ( x ) = t end ( x ) − t as δt ( x ) = − λ ( t ) ln ~φ ( x , t ) · ~φ ( x , t ) N φ ( t ) ! . (27)This finalizes the N component analysis of the time-delay, generalizing the two component (complex) analysisof Ref. [41]. IV. CORRELATION FUNCTIONS
We now derive expressions for the two-point and three-point correlation functions of the time-delay field. To doso, it is convenient to introduce a rescaled version of thecorrelation function ∆( x ) defined through h φ i ( x ) φ j ( ) i = ∆( x ) φ rms δ ij . (28)By definition ∆(0) = 1, and as we vary x , ∆ covers theinterval ∆( x ) ∈ (0 , x ) ∈ (0 , / t end as a func-tion of x, measured in Hubble lengths ( H − ), for variouscombinations of masses. FIG. 3. A plot of the field’s correlation ∆ as a function of x (in units of H − ) for different masses: blue is m = 2 and m ψ = 1 /
2, red is m = 4 and m ψ = 1 /
2, green is m = 2 and m ψ = 1 /
4, orange is m = 4 and m ψ = 1 / A. Two-Point Function
We now express the time-delay correlation functions asa power series in ∆ and 1 / N . An alternative derivationof the power spectra of the time-delay field in terms ofan integral, which is closer to the language of Ref. [41],can be found in Appendix C.Using the above approximation for δt in eq. (27), thetwo-point correlation function for the time-delay is(2 λ ) h δt ( x ) δt ( ) i = * ln ~φ x · ~φ x N φ rms ! ln ~φ · ~φ N φ rms !+ , (29)where we have used the abbreviated notation ~φ x ≡ ~φ ( x ).The two-point function will include a constant (indepen-dent of x ) for a non-zero h δt i . This can be reabsorbedinto a shift in t , whose dependence we have suppressedhere, and so we will ignore the constant in the follow-ing computation. This means that we will compute the connected part of the correlation functions.We would like to form an expansion, but we do nothave a classical trajectory about which to expand. In-stead we use the following idea: we recognize that ~φ · ~φ should be centralized around its mean value of N φ rms ,plus relatively small fluctuations at large N . This meansthat it is convenient to write ~φ · ~φ N φ rms = 1 + ~φ · ~φ N φ rms − ! (30)and treat the term in parenthesis on the right as small, asit represents the fluctuations from the mean. This allows us to Taylor expand the logarithm in powers ofΦ ≡ ~φ · ~φ N φ rms − , (31)with h Φ i = 0. Now recall that the series expansion of thelogarithm for small Φ is ln(1 + Φ) = Φ − Φ + Φ − . . . ,allowing us to expand eq. (29) to any desired order in Φ.The leading non-zero order is quadratic ∼ Φ . It is(2 λ ) h δt ( x ) δt ( ) i = h Φ( x )Φ( ) i , = h φ i ( x ) φ i ( x ) φ j ( ) φ j ( ) i ( N φ rms ) − , (32)where we are implicitly summing over indices in the sec-ond line (for simplicity, we will place all component in-dices ( i, j ) downstairs). Using Wick’s theorem to per-form the four-point contraction, we find the result(2 λ ) h δt ( x ) δt ( ) i = 2∆ ( x ) N , (33)where x ≡ | x | . This provides the leading approximationfor large N , or small ∆. This should be contrasted tothe RSG approximation used in Refs. [21, 22] and sum-marized in [41], in which the correlation function is ap-proximated as ∼ ∆, rather than ∼ ∆ , at leading order.For brevity, we shall not go through the result at eachhigher order here, but we report on results at higher orderin Appendix A. By summing those results to differentorders, we find(2 λ ) h δt ( x ) δt ( ) i = 2∆ N + 2∆ N − N + 16∆ N + 8∆ N − N + 24∆ N + O (cid:18) N (cid:19) , (34)(where ∆ = ∆( x ) here). We find that various cancel-lations have occurred. For example, the − / N termthat enters at cubic order (see Appendix A) has canceled.Note that at a given order in 1 / N there are variouspowers of ∆ . However, by collecting all terms at a givenpower in ∆ , we can identify a pattern in the value of itscoefficients as functions of N . We find(2 λ ) h δt ( x ) δt ( ) i = 2∆ ( x ) N + 2∆ ( x ) N ( N + 2) + 16∆ ( x )3 N ( N + 2)( N + 4)+ 24∆ ( x ) N ( N + 2)( N + 4)( N + 6) + . . . (35)We then identify the entire series as(2 λ ) h δt ( x ) δt ( ) i = ∞ X n =1 C n ( N ) ∆ n ( x ) , (36)where the coefficients are C n ( N ) = 1 n (cid:18) N − nn (cid:19) − , (37)with (cid:0) ab (cid:1) the binomial coefficient. This series is conver-gent for any N and ∆ ∈ (0 , N = 1 we find(2 λ ) h δt ( x ) δt ( ) i = 2∆ ( x ) + 2∆ ( x )3 + 16∆ ( x )45 + . . . = 2 (sin − ∆( x )) . (38)For N = 2 we find(2 λ ) h δt ( x ) δt ( ) i = ∆ ( x ) + ∆ ( x )2 + ∆ ( x )3 + . . . = Li (∆ ( x )) , (39)where Li s ( z ) is the polylogarithm function. We also findthat for any even value of N , the series is given by thepolylogarithm function plus a polynomial in ∆; this isdescribed in Appendix B.Using the power series, we can easily obtain plots of thetwo-point function for any N . For convenience, we plotthe re-scaled quantity N (2 λ ) h δt ( x ) δt ( ) i as a functionof ∆ in Figure 4 (top) for different N . We see convergenceof all curves as we increase N , which confirms that theleading behavior of the (un-scaled) two-point function is ∼ / N . In Figure 4 (bottom) we plot N h δt ( x ) δt ( ) i asa function of x for different masses and two different N . B. Three-Point Function
The three-point function is given by a simple modifi-cation of eq. (29), namely − (2 λ ) h δt ( x ) δt ( y ) δt ( ) i = * ln ~φ x · ~φ x N φ rms ! ln ~φ y · ~φ y N φ rms ! ln ~φ · ~φ N φ rms !+ . (40)Here we will work only to leading non-zero order, whichin this case is cubic. We expand the logarithms as beforeto obtain − (2 λ ) h δt ( x ) δt ( y ) δt ( ) i = h Φ( x )Φ( y )Φ( ) i , = h φ i ( x ) φ i ( x ) φ j ( y ) φ j ( y ) φ k ( ) φ k ( ) i ( N φ rms ) + 2 − (cid:18) h φ i ( x ) φ i ( x ) φ j ( y ) φ j ( y ) i ( N φ rms ) + 2 perms (cid:19) , (41)where “perms” is short for permutations under inter-changing x , y , . Using Wick’s theorem to perform thevarious contractions, we find the result − (2 λ ) h δt ( x ) δt ( y ) δt ( ) i = 8 ∆( | x − y | ) ∆( x ) ∆( y ) N . (42) FIG. 4. Top: a plot of the (re-scaled) two-point function ofthe time-delay N (2 λ ) h δt ( x ) δt ( ) i as a function of ∆ ∈ [0 , N : dot-dashed is N = 1, solid is N = 2, dottedis N = 6, and dashed is N → ∞ . Bottom: a plot of the (re-scaled) two-point function of the time-delay
N h δt ( x ) δt ( ) i as a function of x for different masses: blue is m = 2 and m ψ = 1 /
2, red is m = 4 and m ψ = 1 /
2, green is m = 2 and m ψ = 1 /
4, orange is m = 4 and m ψ = 1 /
4, with solid for N = 2 and dashed for N → ∞ . We would now like to use the three-point functionas a measure of non-Gaussianity. For a single ran-dom variable, a measure of non-Gaussianity is to com-pute a dimensionless ratio of the skewness to the 3/2power of the variance. For a field theory, we symmetrizeover variables, and define the following measure of non-Gaussianity in position space S ≡ h δt ( x ) δt ( y ) δt ( ) i p h δt ( x ) δt ( y ) ih δt ( x ) δt ( ) ih δt ( y ) δt ( ) i . (43)(Recall that we are ignoring h δt i , as it can be just re-absorbed into t , so the three-point and two-point func-tions are the connected pieces). Computing this at theleading order approximation using eqs. (33, 42) (valid forlarge N , or small ∆), we find S ≈ − r N . (44)Curiously, the dependence on x , y has dropped out atthis order. We see that for small N there is significantnon-Gaussianity, while for large N the theory becomesGaussian, as expected from the central limit theorem. C. Momentum Space
Let us now present our results in k -space. We shallcontinue to analyze the results at high N , or small ∆,which allows us to just include the leading order results.For the two-point function, we define the power spec-trum through h δt ( k ) δt ( k ) i = (2 π ) δ ( k + k ) P δt ( k ) . (45)We use eq. (33) and Fourier transform to k -space us-ing the convolution theorem. To do so it is convenientto introduce a dimensionless field ˜ φ i ≡ φ i /φ rms andintroduce the corresponding power spectrum P ˜ φ ( k ) = P φ ( k ) /φ rms = | u k | /φ rms , which is the Fourier trans-form of ∆( x ). We then find the result P δt ( k ) ≈ λ N Z d k ′ (2 π ) P ¯ φ ( k ′ ) P ¯ φ ( | k − k ′ | ) . (46)A dimensionless measure of fluctuations is the following P δt ( k ) ≡ k H P δt ( k )2 π , (47)The factor of k / (2 π ) is appropriate as this gives thevariance per log interval: h ( Hδt ) i = R d ln k P δt ( k ). Bystudying eq. (46), one can show P δt ≈ const for small k and falls off for large k . This creates a spike in P δt ( k )at a finite k ∗ and its amplitude is rather large. (This isto be contrasted with the usual fluctuations in de Sitterspace, P δt ( k ) ∝ /k , making P δt ( k ) approximately scaleinvariant.) We see that the amplitude of the spike scalesas ∼ / N , and so it is reduced for large N . A plot of P δt ( k ) is given in Figure 5.For the three-point function, we define the bispectrumthrough h δt ( k ) δt ( k ) δt ( k ) i = (2 π ) δ ( k + k + k ) B δt ( k , k , k )(48)where we have indicated that the bispectrum only de-pends on the magnitude of the 3 k -vectors, with the con-straint that the vectors sum to zero. We use eq. (42) andFourier transform to k -space, again using the convolutiontheorem. We find B δt ( k , k , k ) ≈ − λ N × (cid:20)Z d k (2 π ) P ¯ φ ( k ) P ¯ φ ( | k − k | ) P ¯ φ ( | k + k | ) + 2 perms (cid:21) (49) FIG. 5. The dimensionless power spectrum
N P δt ( k ) at large N as a function of k (in units of H ) for different choices ofmasses: blue is m = 2 and m ψ = 1 /
2, red is m = 4 and m ψ = 1 /
2, is m = 2 and m ψ = 1 /
4, orange is m = 4 and m ψ = 1 / To measure non-Gaussianity in k -space, it is conven-tional to introduce the dimensionless f NL parameter, de-fined as f NL ( k , k , k ) ≡ B δt ( k , k , k ) P δt ( k ) P δt ( k ) + 2 perms . (50)By substituting the above expressions for P δt and B δt ,we see that f NL is independent of N at this leading or-der. However, this belies the true dependence of non-Gaussianity on the number of fields. This is because f NL is a quantity that can be large even if the non-Gaussianityis relatively small (for example, on CMB scales, any f NL smaller than O (10 ) is a small level of non-Gaussianity).Instead a more appropriate measure of non-Gaussianityin k -space is to compute some ratio of the bispectrumto the 3/2 power of the power spectrum, analogous tothe position space definition in eq. (43). For the simpleequilateral case, k = k = k , we define F NL ( k ) ≡ k / B δt ( k )3 √ πP δt ( k ) / , (51)where we have inserted a factor of k / / ( √ π ) frommeasuring the fluctuations per log interval. Usingeqs. (46, 49), we see that F NL ∝ / √N , as we foundin position space. In Figure 6, we plot this function. Wenote that although the non-Gaussianity can be large, thepeak is on a length scale that is small compared to theCMB and so it evades recent bounds [6]. A factor of 6/5 is often included when studying the gauge invari-ant quantity ζ that appears in cosmological perturbation theory,but it does not concern us here. FIG. 6. The dimensionless bispectrum √N F NL as a functionof k (in units of H ) at large N for m = 4 and m ψ = 1 / V. CONSTRAINTS ON HYBRID MODELS
Hybrid inflation models must satisfy several observa-tional constraints. Here we discuss these constraints, in-cluding the role that N plays, and discuss the implica-tions for the scale of the density spike. A. Topological Defects
The first constraint on hybrid models concerns the pos-sible formation of topological defects. Since the waterfallfield starts at φ = 0 and then falls to some vacuum,it spontaneously breaks a symmetry. For a single field N = 1, this breaks a Z symmetry; see Figure 1. Formultiple fields N >
1, this breaks an SO ( N ) symmetry. N = 1 leads to the formation of domain walls , whichare clearly ruled out observationally, so these models arestrongly disfavored; N = 2 leads to the formation of cos-mic strings , which have not been observed and if theyexist are constrained to be small in number. The sub-ject of cosmic string production in hybrid inflation andtheir subsequent effects is discussed in [23]. For currentbounds on cosmological defects from the Planck collabo-ration the reader can refer to [48] and references therein.Diluting cosmic strings to make them unobservable inour case would require a very large number of e-foldingsof the waterfall phase to make compatible with observa-tions, and seems unrealistic. Further increasing the num-ber of waterfall fields, N = 3 leads to the formation of monopoles , which are somewhat less constrained; N = 4leads to the formation of textures , which are relativelyharmless; N > N >
4, one might be concerned about con- straints from the re-ordering of
N − SO ( N ) global symmetry.However it is important to note that all global symme-tries are only ever approximate. So it is expected thatthese modes are not strictly massless, but pick up a smallmass at some order, as all Goldstones do. The only im-portant point is that the mass of the Goldstones m G issmall compared to the mass scales of the inflaton andthe waterfall field. For example, if we have a scale ofinflation with H ∼ GeV, then the inflaton and wa-terfall fields should have a mass not far from ∼ GeValso. We then only need the Goldstone masses to sat-isfy m G ≪ GeV for our analysis to be true, and forthe SO ( N ) symmetry to be a good approximation forthe purpose of the phase transition. At late times, these(approximate) Goldstones will appear heavy and relax tothe bottom of their potential, and be harmless. So, withthis in mind, a large VEV for the waterfall field is in-deed observationally allowed, and large e-folds after thetransition leading to large black holes is indeed allowed. B. Inflationary Perturbations
Inflation generates fluctuations on large scales whichare being increasingly constrained by data. An impor-tant constraint on any inflation model is the bound on thetensor-to-scalar ratio r . CMB measurements from Planckplace an upper bound on tensor modes of r < .
11 (95%confidence) [6]. The amplitude of tensor modes is directlyset by the energy density during inflation. Typical hy-brid models are at relatively low energy scales, withoutthe need for extreme fine tuning, and so they immediatelysatisfy this bound. Recent data by the BICEP2 experi-ment [42] claim a detection of gravitational waves with r ∼ . V ( ψ ) = V + m ψ ψ . .. ,we have so far considered the regime V ≫ m ψ ψ , whichgenerates a very small amount of tensor modes. Howeverone can operate in a regime with V & m ψ ψ , wherethe tensor to scalar ratio can be pushed to be closer to O (0 . r in hybrid inflation models for future work.Although the detection of tensor modes is not con-firmed , scalar modes are pinned down with great accu-racy. The tilt of the scalar mode spectrum is character-ized by the primordial spectral index n s . WMAP [5] andPlanck measurements [6] place the spectral index near n s,obs ≈ . , (52)giving a red spectrum. Here we examine the constraintsimposed on hybrid models in order to obtain this valueof n s .0The tilt on large scales is determined by the timer field ψ . For low scale models of inflation, such as hybrid in-flation, the prediction for the spectral index is n s = 1 + 2 η, (53)where η ≈ πG V ′′ ψ V = V ′′ ψ H . (54)This is to be evaluated N e e-foldings from the end of in-flation, where N e = 50 −
60 in typical models. Combiningthe above equations, we need to satisfy V ′′ ψ ≈ − . H .If we take V ψ = m ψ ψ /
2, then V ′′ ψ >
0, and n s > ψ where η is evaluated, while leaving the quadraticapproximation for V ψ valid at small ψ . For most reason-able potential functions, such as potentials that flatten atlarge field values, we expect | V ′′ ψ | . m ψ . So this suggestsa bound m ψ & . H , (55)which can only be avoided by significant fine tuning ofthe potential. Hence although the timer field is assumedlight ( m ψ < H ), it cannot be extremely light.As an example let us consider a potential that canbe approximated by V ( ψ ) = m ψ ψ − g ψ . We areneglecting higher order terms needed to stabilize the po-tential. Let us take V ′′ ( ψ CMB ) = m ψ − g ψ < φ CMB is the field value at the point where the CMBfluctuations exit the horizon during inflaiton, in order toproduce a red-tilted spectrum. This requirement leads to m ψ . g ψ which in turns implies V ′′ ( ψ CMB ) ∼ m ψ .Since fluctuations that are imprinted on the CMB mustexit the horizon sufficiently before the waterfall transi-tion, in order for the CMB spectrum to keep its approx-imate scale-invariance, the quartic term in the potentialconsidered in this example will be subdominant duringthe waterfall transition, since ψ waterfall < ψ CMB . An ex-ample potential for the timer field is V ψ ( ψ ) = F m ψ ψ /F ) . (56)If one Taylor expands this, one finds the needed m ψ ψ / concave down , which is compatiblewith the observed red-tilted spectrum.Since a detailed calculation of the primordial black holeproduction for specific realizations of our model (alongwith connecting it to potential observables like super-massive black holes) is deferred for a future presentation,the corresponding choice of potential parameters will bedone at that time as well. C. Implications for Scale of Spike
The length scale associated with the spike in the spec-trum is set by the Hubble length during inflation H − ,red-shifted by the number of e-foldings of the waterfallphase N w . Since the Hubble scale during inflation is typ-ically microscopic, we need the duration of the waterfallphase N w to be significant (e.g., N w ∼ −
40) to obtaina spike on astrophysically large scales. Here we examineif this is possible.Since we have defined t = 0 to be when the transitionoccurs ( ψ = ψ c ), then N w = Ht with final value at t = t end . To determine the final value, we note that modesgrow at the rate λ , derived earlier in eq. (22). For m ψ 40) the exponential factorexp( − p t ) ≈ exp − m ψ N w H ! (57)is somewhat small and we will ignore it here. In thisregime, the dimensionless growth rate λ/H can be ap-proximated as a constant λH ≈ − 32 + r 94 + m H . (58)The typical starting value for φ is roughly of order H (de Sitter fluctuations) and the typical end value for φ isroughly of order M P l (Planck scale). For self consistency, φ must pass from its starting value to its end value in N w e-foldings with rate set by λ/H . This gives the approxi-mate value for N w as N w ≈ Hλ ln (cid:18) M P l H (cid:19) . (59)This has a clear consequence: If we choose m ≫ H , as isdone in some models of hybrid inflation, then H/λ ≪ H to be many, many orders of mag-nitude smaller then M P l , then N w will be rather small.This will lead to a spike in the spectrum on rather smallscales and possibly ignorable to astrophysics.Note that if we had ignored the spectral index boundthat leads to eq. (55), then we could have taken m ψ ar-bitrarily small, leading to an arbitrarily small p value. Inthis (unrealistic) limit, it is simple to show λH ≈ m ψ m N w H . (60) A better approximation comes from tracking the full time de-pendence of λ and integrating the argument of the exponentialexp( R t dt ′ λ ( t ′ )), but this approximation suffices for the presentdiscussion. m ψ arbitrarily small, λ could be made small,and N w could easily be made large. However, the exis-tence of the spectral index bound essentially forbids this,requiring us to go in a different direction.The only way to increase N w and satisfy the spectralindex bound on m ψ is to take m somewhat close to H .This allows H/λ to be appreciable from eq. (58). Forinstance, if we set m = 1 . H , then H/λ ≈ 2. If wethen take H just a few orders of magnitude below M P l ,say H ∼ − , M P l , which is reasonable for inflationmodels, we can achieve a significant value for N w . Thiswill lead to a spike in the spectrum on astrophysicallylarge scales, which is potentially quite interesting. It ispossible that there will be distortions in the spectrumby taking m close to H , but we will not explore thosedetails here. However, there is an important consequencethat we explore in the next subsection. D. Eternal Inflation Since we are being pushed towards a somewhat lowvalue of m , near H , we need to check if the theory stillmakes sense. One potential problem is that the theorymay enter a regime of eternal inflation. This could occurfor the waterfall field at the hill-top. This would wipeout information of the timer field, which established theapproximately scale invariant spectrum on cosmologicalscales.The boundary for eternal inflation is roughly when thedensity fluctuations are O (1), and this occurs when thefluctuations in the time delay are h ( H δt ) i = O (1). Toconvert this into a lower bound on m , let us imagine that m is even smaller than H . In this regime, the growthrate λ can be estimated using eq. (58) as λ ∼ m /H .Using eq. (33) this gives h ( H δt ) i ∼ H / ( N m ). Thisimplies that eternal inflation occurs when the waterfallmass is below a critical value m c , which is roughly m c ∼ H N / . (61)So when N ∼ m near H , becausewe then enter eternal inflation. On the other hand, forlarge N we are allowed to have m near H and avoidthis problem. This makes sense intuitively, because formany fields it is statistically favorable for at least one ofthe fields to fall off the hill-top, causing inflation to end.Hence large N is more easily compatible with the aboveset of constraints than low N . VI. DISCUSSION AND CONCLUSIONS In this work we studied density perturbations in hybridinflation caused by N waterfall fields, which contains aspike in the spectrum. We derived expressions for cor-relation functions of the time-delay and constrained pa-rameters with observations. Density Perturbations : We derived a convergent seriesexpansion in powers of 1 / N and ∆( x ), the dimensionlesscorrelation function for the field, for the two-point func-tion of the time-delay for any N , and the leading orderbehavior of the three-point function of the time-delay forlarge N . These correlation functions are well approxi-mated by the first term in the series for large N (evenfor N = 2 the leading term is moderately accurate to ∼ N ). In thisregime, the fluctuations are suppressed, with two-pointand three-point functions given by h δt ( x ) δt ( ) i ≈ ∆ ( x )2 λ N , h δt ( x ) δt ( y ) δt ( ) i ≈ − ∆( | x − y | )∆( x )∆( y ) λ N . (62)Although this reduces the spike in the spectrum, for anymoderate value of N , such as N = 3 , , , the amplitudeof the spike is still quite large (orders of magnitude largerthan the ∼ − level fluctuations on larger scales rele-vant to the CMB), and may have significant astrophysicalconsequences. Also, the relative size of the three-pointfunction to the 3/2 power of the two-point function scalesas ∼ / √N . In accordance with the central limit theo-rem, the fluctuations become more Gaussian at large N .This will make the analysis of the subsequent cosmologi-cal evolution more manageable, as this provides a simplespectrum for initial conditions. We note that since weare considering small scales compared to the CMB, thenthis non-Gaussianity evades Planck bounds [6]. Constraints : We mentioned that hybrid models avoidtopological defects for large N , while tensor mode con-straints can be satisfied in different models. A very seri-ous constraint on hybrid models comes from the observedspectral index n s ≈ . 96, which requires the potential toflatten at large field values. One consequence of this isthat the timer field mass m ψ needs to be only a littlesmaller than the Hubble parameter during inflation H ,or else the model is significantly fine tuned. For a largevalue of the waterfall field mass m , this would implya large growth rate of fluctuations, a rapid terminationof inflation, and in turn a density spike on very smallscales. Otherwise, we need to make the waterfall fieldmass m somewhat close to H , but this faces problemswith eternal inflation. However, by using a large numberof waterfall fields N , it is safer to make the waterfall fieldmass m somewhat close to H . This reduces the growthrate of fluctuations, prolonging the waterfall phase formany e-foldings.Thus large N presents a plausible setup to estab-lish a spike in the density perturbations on astro-physically large length scales that is consistent withother constraints. Outlook : It may be possible that these perturbationsseed primordial black holes, which may be relevant toseeding supermassive black holes, or an intriguing form of2dark matter. Since black hole formation is exponentiallysensitive on the inflationary power spectrum, an accuratecalculation of fluctuations is important, but predictingastrophysical observables does not follow trivially. Aninvestigation into these topics is underway. It would beimportant to fully explore the eternal inflation bound andthe effects on the spectrum for relatively light waterfallfield masses. Finally, it would be of interest to try toembed these large N models into fundamental physics,such as string theory, and to explore reheating [51–54]and baryogenesis [55–61] in this framework. Acknowledgments We would very much like to thank Alan Guth for hisguidance and contribution to this project. We wouldalso like to thank Victor Buza and Alexis Giguere forvery helpful discussions. This work is supported by theU.S. Department of Energy under grant Contract Num-ber de-sc00012567 and in part by MIT’s Undergradu-ate Research Opportunities Program. EIS gratefully ac-knowledges support from a Fortner Fellowship at UIUC.Research at Perimeter Institute is supported by the Gov-ernment of Canada through Industry Canada and by theProvince of Ontario through the Ministry of Researchand Innovation. Appendix A: Series Expansion to Higher Orders Earlier we computed the two-point correlation functionfor the time-delay at quadratic order, and then statingthe results at all orders. Here we mention the resultsorder by order. a. Cubic Order At cubic order ∼ Φ we find(2 λ ) h δt ( x ) δt ( ) i = − N (A1) b. Quartic Order At quartic order ∼ Φ we find(2 λ ) h δt ( x ) δt ( ) i = 8∆ + 2∆ N + 40∆ + 4∆ N (A2) c. Quintic Order At quintic order ∼ Φ we find(2 λ ) h δt ( x ) δt ( ) i = − + 32∆ N − + 64∆ N (A3) d. Sextic Order At sextic order ∼ Φ we find(2 λ ) h δt ( x ) δτ ( ) i = 168∆ + 72∆ + 16∆ N + 1056∆ + 464∆ + 32∆ N + 6144∆ + 2496∆ + 128∆ N (A4) e. Septic Order At septic order ∼ Φ we find(2 λ ) h δt ( x ) δt ( ) i = − + 22∆ + 6∆ ) N + O (cid:18) N (cid:19) (A5) f. Octic Order At octic order ∼ Φ we find(2 λ ) h δt ( x ) δt ( ) i = 8(72∆ + 39∆ + 16∆ + 3∆ ) N + O (cid:18) N (cid:19) (A6) Appendix B: Two-Point Function for Even Numberof Fields When N is even the expression always involves thepolylog function that we found for N = 2, plus correc-tions that depend on N . We find that the form of theanswer is(2 λ ) h δt ( x ) δt ( ) i = Li (∆ ) + P N (∆ )∆ N − + ¯ P N (∆ ) ln(1 − ∆ )∆ N − , (B1)where P N (∆ ) is a polynomial of degree ( N − / , (B2)¯ P N (∆ ) is a polynomial of degree ( N − / . (B3)For N = 4, we find P (∆ ) = − , (B4)¯ P (∆ ) = − . (B5)For N = 6, we find P (∆ ) = 12 − 74 ∆ , (B6)¯ P (∆ ) = 12 − + 32 ∆ . (B7)3For N = 8, we find P (∆ ) = − 13 + 43 ∆ − , (B8)¯ P (∆ ) = − 13 + 32 ∆ − + 116 ∆ . (B9) Appendix C: Alternative Derivation of Time-DelaySpectra Here we write the two-point function as a multidimen-sional integral. It is convenient now to switch to a vectornotation thus making the components of φ explicit.˜ φ ( , t ) ≡ ~φ x ≡ ( X , X , X , . . . , X N ) , (C1)˜ φ ( x , t ) ≡ ~φ ≡ ( X N +1 , X N +2 , X N +3 , . . . , X N ) , (C2) ~X ≡ ( X , X , X , . . . , X N ) . (C3)The average value of a function F of a random variable ~X with probability distribution function p ( X ) is given by h F [ X ] i = Z dXp ( X ) F [ X ] . 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