A density spike on astrophysical scales from an N-field waterfall transition

(1)

A density spike on astrophysical scales

from an N-field waterfall transition

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Citation

Halpern, Illan F., Mark P. Hertzberg, Matthew A. Joss, and

Evangelos I. Sfakianakis. “A Density Spike on Astrophysical Scales

from an N-Field Waterfall Transition.” Physics Letters B 748

(September 2015): 132–143.

As Published

http://dx.doi.org/10.1016/j.physletb.2015.06.076

Publisher

Elsevier

Version

Final published version

Citable link

http://hdl.handle.net/1721.1/98205

Terms of Use

Creative Commons Attribution

(2)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

A

density

spike

on

astrophysical

scales

from

an

N

-ﬁeld

waterfall

transition

Illan

F. Halpern

a

,

b

,

Mark

P. Hertzberg

a

,

∗

,

Matthew

A. Joss

a

,

Evangelos

I. Sfakianakis

a

,

c a_Center_for_Theoretical_Physics_and_Dept._of_Physics,_{Massachusetts}_Institute_of_Technology,_Cambridge,_MA_02139,_USA

b_Perimeter_Institute_for_Theoretical_Physics,_Waterloo,_ON_N2L_2Y5,_Canada c_Dept._of_Physics,_University_of_Illinois_at_{Urbana-Champaign,}_Urbana,_IL_61801,_USA

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received16October2014

Receivedinrevisedform25June2015 Accepted30June2015

Availableonline2July2015 Editor:S.Dodelson

Hybrid inflation modelsare especially interestingasthey leadto aspike inthe densitypower spec-trum on smallscales, compared tothe CMB,while alsosatisfying currentbounds ontensor modes. HerewestudyhybridinflationwithN waterfallfieldssharingaglobalSO(N )symmetry.Theinclusion ofmanywaterfallfieldshasthe obviousadvantage ofavoidingtopologicallystable defectsfor N >3. We findthat it also hasanother advantage: it is easier to engineermodels that can simultaneously (i) becompatiblewithconstraintsonthe primordialspectralindex, whichtendstootherwisedisfavor hybrid models,and(ii)produceaspikeonastrophysicallylarge lengthscales.Thelattermayhave sig-nificantconsequences,possiblyseedingtheformationofastrophysicallylarge blackholes.Wecalculate correlation functionsofthe time-delay,ameasureofdensityperturbations,producedby thewaterfall fields, as aconvergent power seriesinboth1/N andthe field’s correlation function(x).We show that for large N,the two-pointfunction isδt(x)δt(0)∝ 2(|x|)/N and thethree-point functionis

δt(x)δt(y)δt(0)∝ (|x−y|)(|x|)(|y|)/N2_._In_accordance_with_the_central_limit_theorem,_the_density

perturbationsonthescaleofthespikeareGaussianforlarge_N andnon-Gaussianforsmall_N.

1. Introduction

Inflation, a phase of accelerated expansion in the very early universe thought to be driven by one or severalscalar fields,is our paradigm of early universe cosmology [1–4]. It naturally ex-plains the large scale homogeneity, isotropy, and flatness of the universe.Moreover,thebasicpredictionsofeventhesimplest sin-glefieldslowrollmodels,givingapproximatescaleinvarianceand smallnon-Gaussianityinthe

∼

10−5 _level_departures_from homo-geneityandisotropy,are inexcellentagreementwithrecentCMB data[5–7]andlargescalestructure.

While the basic paradigm of inﬂation is in excellent shape, no single model stands clearly preferred. Instead the literature abounds with various models motivated by different considera-tions,suchasstringmoduli,supergravity,branes,ghosts,Standard Model,etc.[8–19].Whiletheincomingdataisatsuchan impres-sivelevelthatitcandiscriminatebetweenvariousmodelsandrule

*

Correspondingauthor.

E-mailaddresses:ihalpern@perimeterinstitute.ca(I.F. Halpern),mphertz@mit.edu

(M.P. Hertzberg),mattjoss@mit.edu(M.A. Joss),esfaki@illinois.edu(E.I. Sfakianakis).

out many,such asmodels that overpredict non-Gaussianity, it is notclearifthedatawilleverrevealonemodelalone.Animportant way tomake progress isto disfavormodels based on theoretical grounds(suchasissuesofunitarityviolation, acausality,etc.)and toﬁndamodelthatisabletoaccountforphenomenainthe uni-verselacking analternateexplanation.Itisconceivablethatsome version oftheso-called“hybrid inﬂation”modelmayaccountfor astrophysicalphenomena,forreasonsweshallcometo.

The hybrid inflation model,originally proposed by Linde [20], requiresatleasttwo fields.Oneofthefieldsislight andanother ofthe fieldsisheavy(in Hubbleunits).The lightfield, calledthe “timer”, isatearly timesslowly rollingdown apotential hilland generates thealmost scale invariant spectrum offluctuations ob-served in the CMB andin large scale structure. The heavy field, called the “waterfall” field, has an effective mass that is time-dependent and controlled by the value of the timer field. The waterfallfield isoriginallytrappedataminimumofitspotential, butasitseffectivemass-squaredbecomesnegative,atachyonic in-stability follows,leading to theendof inflation;an illustrationis given in

Fig. 1

. The name“hybrid inflation”comes from the fact that this model is a sort of hybrid between a chaotic inflation modelandasymmetrybreakinginflationmodel.

http://dx.doi.org/10.1016/j.physletb.2015.06.076

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Fig. 1. Anillustrationoftheevolutionoftheeffectivepotential forthe waterfall ﬁeldφasthetimerﬁeldψevolvesfrom“high”valuesatearlytimes,toψ= ψc,

andﬁnallyto“low”valuesatlatetimes.Intheprocess,theeffectivemass-squared ofφevolvesfrompositive,tozero,tonegative(tachyonic).

As originally discussed in Refs. [21,22], one of the most fas-cinating features of hybrid models is that the tachyonic behav-ior of the waterfall ﬁeld leads to a sharp “spike” in the density powerspectrum. Thiscould seed primordial blackholes [23–28]. Forgenericparameters,thelengthscaleassociatedwiththisspike is typically very small. However, if one could ﬁnd a parameter regime where the waterfall phase were to be prolonged, lasting formany e-foldings, say Nw

∼

30–40,then thiswould lead to a

spikein the densityperturbations on astrophysically large scales (butsmallerthanCMBscales).Thismayhelptoaccount for phe-nomenasuch assupermassiveblackholes ordarkmatter, etc.Of course the details of all thisrequires a very carefulexamination ofthe spectrum ofdensityperturbations, including observational constraints.

Ordinarily the spectrum of density perturbations in a given modelof inflation is obtained by decomposing the inflatonfield into a homogeneous part plus a small inhomogeneous perturba-tion.However,forthe waterfallfieldsofhybridinflation, this ap-proachfailssinceclassicallythewaterfallfield wouldstayforever atthetop ofaridge inits potential.It isthe quantum perturba-tions themselves that lead to a non-trivial evolution of the wa-terfall field, and therefore the quantum perturbations cannot be treatedas small.Several approximationshave been used to deal with this problem [21,22,29–40]. Here we follow the approach presented by one of usrecently in Ref. [41], where a free field time-delaymethodwasused,providingaccuratenumericalresults. Inthispaper,wegeneralizethemethodofRef.[41]toamodel with

N

waterfallﬁeldssharingaglobalSO

(N )

symmetry.Amodel ofmanyﬁeldsmaybenaturalinvariousmicroscopicconstructions, such asgranduniﬁed models, stringmodels, etc. But apartfrom generalizingRef.[41]to

N

ﬁelds,wealsogomuchfurtherinour analysis:wederiveexplicitanalyticalresultsforseveralcorrelation functionsofthe so-calledtime-delay.We formulate a convergent seriesexpansioninpowersof1

/

N

andtheﬁeld’scorrelation func-tion

(

x

)

.Weﬁndalltermsintheseriestoobtainthetwo-point correlationfunction ofthe time-delayforany

N

.Wealso obtain the leading order behavior at large

N

forthe three-point func-tiontime-delay,whichprovidesameasureofnon-Gaussianity.We ﬁndthatthenon-Gaussianityisappreciableforsmall

N

and sup-pressedforlarge

N

.

Wealsoanalyzeindetail constraintsonhybridinﬂation mod-els.Wecommentonhowmultipleﬁeldsavoid topologicaldefects, whichisaseriousproblemforlow

N

models.However,themost severeconstraintonhybridmodelscomesfromtherequirementto obtainthe observedspectral indexns.We show that atlarge

N

,

it iseasiertoengineermodelsthat canﬁttheobserved ns,while

also allowing for a prolonged waterfall phase. This means that large

N

models provide the most plausible way for the spike to appearon astrophysically large scales andbe compatiblewith otherconstraints.

Withregards to tensor modes, theconfirmation ofthe recent detectionandamplitudeofprimordialgravitationalwaves[42] de-pends largely on understanding dust foregrounds [43]. The final answer to theexistence of primordialtensor modeswill have to begivenbyfutureexperiments,sincearecentjointanalysisof cur-rentdata[44]doesnotprovideacleardetection.Sowe willonly briefly discusspossibilities ofobtaining observable tensor modes fromhybridinflationinSection5.

The organization of our paper is as follows: In Section 2 we present our hybrid inﬂation model and discuss our approxima-tions.InSection 3we presentthetime-delayformalism,adapting themethodofRef. [41] to

N

ﬁelds.InSection4 wederive a se-ries expansion for the two-point function, we derive the leading orderbehavior ofthe three-point function,andwe derive results ink-space.InSection5we presentconstraintson hybridmodels, emphasizing the role that

N

plays. In Section 6 we discussand conclude. Finally,intheAppendiceswe presentfurtheranalytical results.

2.

N

ﬁeldmodel

Themodelconsistsoftwotypesofﬁelds:Thetimerﬁeld

ψ

that drives thefirstslow-rollinflation phase,andthewaterfallfield

φ

thatbecomes tachyonicduring thesecondphasecausinginﬂation toend.Inmanyhybridmodels,

φ

iscomprisedoftwocomponents, acomplexﬁeld,buthereweallowfor

N

realcomponents

φ

i.We

assumethecomponentsshareaglobalSO

(N )

symmetry,andsoit isconvenienttoorganizethemintoavector

φ = {φ

1

, φ

2

, . . . , φ

N

}.

(1)

Forthespecialcase

N =

2,thiscanbe organizedintoa complex ﬁeldbywriting

φ

complex

= (φ

1

+

i

φ

2

)/

√

2.

Thedynamicsisgovernedbythestandardtwoderivativeaction forthescalarﬁelds

ψ

,

φ

minimally coupledtogravity asfollows (signature

+−−−

) S

=

d4x

√

−

g

1 16

π

GR

+

1 2g μν

_∂

μ

ψ ∂

ν

ψ

+

1 2g μν

_∂

μ

φ · ∂

ν

φ −

V

(ψ, φ)

.

(2)

The potential V is given by a sum of terms: V0 providing false vacuumenergy,Vψ

(ψ)

governingthetimerﬁeld,Vφ

(φ)

governing

thewaterfallﬁeld, andVint

(ψ,

φ)

governingtheir mutual

interac-tion,i.e.,

V

(ψ, φ)

=

V0

+

Vψ

(ψ )

+

Vφ

(φ)

+

Vint

(ψ, φ).

(3)

During inﬂation we assume that the constant V0 dominates all otherterms.

The timer ﬁeld potential Vψ and the waterfall ﬁeld potential

Vφ caningeneralbe complicated.Ingeneral,they areallowedto

benon-polynomial functionsaspartofsomelowenergyeffective ﬁeld theory,possiblyfromsupergravityorother microscopic the-ories. Forour purposes, it is enough to assume an extremumat

ψ

= φ =

0 andexpandthepotentialsaroundthisextremumas fol-lows:

(4)

Vψ

(ψ )

=

1 2m 2 ψ

ψ

2

+ . . .

(4) Vφ

(φ)

= −

1 2m 2 0

φ · φ + ...

(5)

The timerﬁeld is assumedtobe light mψ

<

H andthe waterfall

ﬁeldisassumedtobeheavy m0

>

H ,where H istheHubble pa-rameter.Intheoriginal hybridmodel,thisquadratictermfor Vψ

wasassumedtobetheentirepotential.Thismodelleadstoa spec-trumwithaspectralindexns

>

1 andisobservationallyruledout.

Insteadwe needhigherorder terms,indicatedby thedots, toﬁx thisproblem.Thisalsoplacesconstraintsonthemassofthetimer ﬁeldmψ,whichhasimportantconsequences.Wediscussthese

is-suesindetailinSection5.

As indicated by thenegative mass-squared, thewaterfall ﬁeld istachyonicaround

φ

= ψ =

0.Thisobviouslycannotbetheentire potentialbecausethenthepotentialwouldbeunboundedfrom be-low.Instead there mustbe higherorder terms that stabilizethe potential with a global minimum near V

≈

0 (effectively setting thelate-timecosmologicalconstant).

Thekey tohybridinﬂation istheinteractionbetweenthetwo ﬁelds.Forsimplicity,weassumeastandarddimension 4coupling oftheform

Vint

(ψ, φ)

=

1 2g

2

_ψ

2

_{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 fromzero,andthentobecometachyoniconcethetimerﬁeld ap-proachestheorigin;thisisillustratedin

Fig. 1

.

2.1. Approximations

Weassume thatthe constant V0 isdominantduring inﬂation, leading to an approximatede Sitter phase withconstant Hubble parameterH .ByassumingaﬂatFRWbackground,thescalefactor isapproximatedas

a

=

exp

(

Ht

).

(7)

At early times,

ψ

is displaced from its origin, so

φ

=

0. This meansthat we canapproximatethe

ψ

dynamicsby ignoringthe backreactionfrom

φ

.The ﬂuctuationsin

ψ

establishnearlyscale invariantﬂuctuationsonlarge scales,whichwe shallreturntoin Section5.However,forthepresentpurposesitisenoughtotreat

ψ

asa classical, homogeneous ﬁeld

ψ(

t

)

.We make the approxi-mationthatwecanneglectthehigherordertermsinthepotential

Vψ inthetransitionera,leadingtotheequationofmotion

¨ψ +

3H

˙ψ +

m2_ψ

ψ

=

0

.

(8)

Solving this equation for

ψ(

t

)

, we insert this into the equation for

φ

i. We allow spatial dependence in

φ

i, and ignore, for

sim-plicity, the higherorder terms in Vφ, leading to the equation of

motion

¨φ

i

+

3H

˙φ

i

−

e−2Ht

∇

2

φ

i

+

m2φ

(

t

)φ

i

=

0

.

(9)

Herewehaveidentiﬁedaneffectivemass-squaredforthewaterfall ﬁeldof m2_φ

(

t

)

≡ −

m2₀

1

−

ψ (

t

)

ψ

c

2

,

(10)

wherethedimension4couplingg hasbeentradedforthe param-eter

ψ

c as g2

=

m20

/ψ

c2. The quantity

ψ

c has the physical

inter-pretationasthe “critical”valueof

ψ

such thatthe effectivemass

of the waterfall ﬁeld passes through zero. So at early times for

ψ > ψ

c, then m2_φ

>

0 and

φ

i is trapped at

φ

i

=

0, while at late

timesfor

ψ < ψ

c,thenm2φ

<

0 and

φ

i istachyonicandcan grow

inamplitude,dependingonthemodeofinterest;

Fig. 1

illustrates thesefeatures.

2.2. Modefunctions

Since we ignore the back-reactionof

φ

onto

ψ

andsince we treat

ψ

ashomogeneousintheequationofmotionfor

φ

(eq.(9)), thenbypassingtok-space,allmodesaredecoupled.Eachwaterfall ﬁeld

φ

i canbe quantizedandexpanded inmodesin momentum

spaceasfollows

φ

i

(

x

,

t

)

=

_d3_k

(

2

π

)

3

ck,ieik·xuk

(

t

)

+

c†_k_,_ie−ik·xuk∗

(

t

)

,

(11)

where c†_k

(

ck

)

are the creation (annihilation) operators, acting on

the

φ

i

=

0 vacuum. Byassuming an initialBunch–Daviesvacuum

for each

φ

i, the mode functions uk are the same for all

com-ponents dueto the SO

(N )

symmetry. To be precise, we assume that atasymptotically earlytimesthe modefunctionsarethe or-dinaryMinkowskispacemodefunctions, withthecaveatthat we needtoinsertfactorsofthescalefactora toconvertfromphysical wavenumberstocomovingwavenumbers,i.e.,

uk

(

t

)

→

e−i k t/a

a

√

2k

,

at early times

.

(12)

Atlatetimesthemodefunctionsbehaveverydifferently.Sincethe ﬁeld becomes tachyonic, the mode functionsgrow exponentially at late times.The transition depends on the wavenumber of in-terest. Thefulldetailsofthemode functionswere explainedvery carefullyinRef.[41],andtheinterested readerisdirected tothat paperformoreinformation.

Since we are approximating

φ

i as a free ﬁeld here, its

ﬂuc-tuations are entirely Gaussian and characterized entirely by its equal time two-point correlation function

φ

i

(

x

)

φ

j

(

y

)

. Passing

to k-space, andusingstatisticalisotropy andhomogeneityof the Bunch–Daviesvacuum,theﬂuctuationsareequallywell character-izedbytheso-calledpowerspectrumPφ

(

k

)

,deﬁnedthrough

φ

i

(

k1

) φ

j

(

k2

)

= (

2

π

)

3

δ(

k1

+

k2

) δ

i jPφ

(

k1

).

(13)

Thismeansthepowerspectrumis

δ

i jPφ

(

k

)

=

d3x e−ik·x

φ

i

(

x

)φ

j

(

0

)

.

(14)

It is simple to show that the power spectrum is related to the modefunctionsby

Pφ

(

k

)

= |

uk

|

2

.

(15)

In Fig. 2 we plot a rescaled version of Pφ, wherewe divideout

by theroot-mean-square(rms) of

φ

deﬁnedas

φ

_rms2

= φ

_i2

(

0

)

.As willbementionedinthenextsection andisextensivelydiscussed in[41],theratio

˜φ(

t

)

=

φ (

t

)

φ

rms

(

t

)

(16) istime-independentforlatetimes,hencesois P_˜φ

(

k

)

.

(5)

Fig. 2. A plot of the (re-scaled) ﬁeld’s power spectrum P_˜φ as a function of wavenumberk (inunitsofH )fordifferentmasses:blueism0=2 andmψ=1/2, redism0=4 andmψ=1/2,greenism0=2 andmψ=1/4,orangeism0=4 and mψ=1/4.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,the readerisreferredtothewebversionofthisarticle.)

3. Thetime-delay

Wewould now like torelate theﬂuctuations inthe waterfall ﬁeld

φ

toaﬂuctuationinaphysicalobservable,namelythedensity perturbation.Animportantstepinthisdirectionistocomputethe so-called“time-delay”

δ

t

(

x

)

; the time offsetforthe endof inﬂa-tionfordifferentpartsoftheuniverse.Thiscausesdifferentregions oftheuniverseto haveinﬂated morethan others,creatinga dif-ference intheir densities (though we willnot explicitly compute

δ

ρ

/

ρ

here). This basicformalism was ﬁrst introduced by Hawk-ing[45]andby GuthandPi[46],andhasrecentlybeenreviewed inRef. [47], wherethe transition fromthe time-delay formalism tothemorefrequentlyusedcurvatureperturbation

R

isoutlined. Inthecontext ofhybrid inﬂation, itwas recentlyused byone of usinRef.[41].Itprovidesanintuitive andstraightforwardwayto calculateprimordial perturbations andwe now usethis to study perturbationsestablishedbythe

N

waterfallﬁelds.

Inits original formulation, the time-delayformalism starts by considering a classical homogeneous trajectory

φ

0

= φ

0

(

t

)

, and then considers a first orderperturbation around this. At first or-der,oneisabletoprovethatthefluctuatinginhomogeneousfield

φ (

x

,

t

)

isrelatedtotheclassicalﬁeld

φ

0,uptoanoveralltime off-set

δ

t

(

x

)

,

φ (

x

,

t

)

= φ

0

(

t

− δ

t

(

x

)).

(17)

In the present case of hybrid inﬂation, the waterfall ﬁeld is initiallytrapped at

φ

=

0 and then onceit becomes tachyonic, it eventually fallsoffthehill-top dueto quantumﬂuctuations. This meansthat thereisnoclassicaltrajectoryaboutwhichtoexpand. Nevertheless,wewillshowthat,toagoodapproximation,theﬁeld

φ (

x

,

t

)

iswelldescribedbyanequationoftheform

(17)

,fora suit-ablydeﬁned

φ

0.Thekeyisthat allmodesofinterestgrowatthe samerateatlatetimes.Furtherinformationofthetimeevolution ofthemodefunctionscanbefoundinRef.[41].

Toshowthis, weneedtocomputetheevolutionoftheﬁeld

φ

accordingto eq. (9). This requiresknowing mφ

(

t

)

, which inturn

requiresknowing

ψ(

t

)

. Bysolving eq. (8)forthe timer ﬁeld and dispensingwithtransientbehavior,wehave

ψ (

t

)

= ψ

cexp

(

−

p t

) ,

(18) where p

=

H

⎛

⎝

3 2

−

9 4

−

m2_ψ H2

⎞

⎠

(19)

(note p

>

0). We have set the origin of time t

=

0 to be when

ψ

= ψ

c andassume

ψ > ψ

c atearlytimes.

Substitutingthissolutionintomφ

(

t

)

,wecan,inprinciple,solve

eq. (9).In general, thesolution is somewhat complicated witha non-trivialdependenceonwavenumber.However,atlatetimesthe behaviorsimpliﬁes.Ourmodesofinterestaresuper-horizonatlate times. For these modes, the gradient term is negligible and the equationofmotionreducesto

¨φ

i

+

3H

˙φ

i

+

mφ2

(

t

)φ

i

=

0

.

(20)

So each mode evolves in the same way at late times. Treating

mφ

(

t

)

asslowlyvarying(whichisjustiﬁedbecausethetimerﬁeld

massmψ

<

H andsop issmall),wecansolvefor

φ

i atlatetimes

t intheadiabaticapproximation.Weobtain

φ

i

(

x

,

t

)

= φ

i

(

x

,

t0

)

exp

⎛

⎝

t

t0 dt

λ(

t

)

⎞

⎠ ,

(21) where

λ(

t

)

=

H

⎛

⎝−

3 2

+

9 4

+

m2₀ H2

1

−

e−2 p t

⎞

⎠ .

(22)

Heret0 issomereferencetime.Fort

>

0,wehave

λ(

t

)

>

0,sothe modesgrow exponentiallyin time. LaterinSection 5we explain thatinfact

λ

isroughlyconstantinthelatterstageofthewaterfall phase,i.e.,theexp

(

−

2pt

)

piecebecomessmall.

We now discuss fluctuations in the time at which inflation ends.Forconvenience, we define thereferencetime t0 to be the time atwhichtherms valueofthefield reaches

φ

end;theendof

inﬂation

N

φ

2rms

(

t0

)

= φ

end2

,

(23)

where we have included a factor of

N

to account for all ﬁelds, allowing

φ

rmstorefertoﬂuctuationsinasinglecomponent

φ

i,i.e.,

φ

2

rms

= φ

i2

(

0

)

.Intermsofthepowerspectrum,itis

φ

_rms2

=

d3k

(

2

π

)

3Pφ

(

k

).

(24)

If we were to include arbitrarily high k, this would diverge quadratically, which is the usual Minkowski space divergence. However, our present analysisonly applies to modesthat are in thegrowingregime.Forthesemodes, Pφ

(

k

)

fallsoffexponentially

withk andthereisnoproblem.Sointhisintegral,weonlyinclude modesthatareintheasymptoticregime,or,roughlyspeaking,only super-horizonmodes.

Using eq. (21), we can express the ﬁeld

φ

i

(

x

,

t

)

at time t

=

t0

+ δ

t intermsoftheﬁeld

φ

i

(

x

,

t0

)

by

φ

i

(

x

,

t

)

= φ

i

(

x

,

t0

)

exp

(λ(

t0

) δ

t

) .

(25)

Ift ischosentobethetimetend

(

x

)

atwhichinﬂationendsateach point inspace, then

φ· φ

x

,

tend

(

x

)

= φ

2

end

=

N φ

2

rms

(

t0

)

, andthe aboveequationbecomes

N

φ

2_rms

(

t0

)

= φ(

x

,

t0

)

· φ(

x

,

t0

)

exp

(

2

λ(

t0

) δ

t

),

(26) whichcanbesolvedforthetime-delayﬁeld

δ

t

(

x

)

=

tend

(

x

)

−

t0as

(6)

Fig. 3. Aplotoftheﬁeld’scorrelationasafunctionofx (inunitsofH−1₎_for differentmasses:blueism0=2 andmψ=1/2,redism0=4 andmψ=1/2,green ism0=2 andmψ=1/4,orangeism0=4 andmψ=1/4.(Forinterpretationofthe referencestocolorinthisﬁgurelegend,thereaderisreferredtothewebversionof thisarticle.)

δ

t

(

x

)

=

−

1 2

λ(

t0

)

ln

φ(

x

,

t0

)

· φ(

x

,

t0

)

N

φ

2rms

(

t0

)

.

(27)

Thisﬁnalizesthe

N

componentanalysisofthetime-delay, gener-alizingthetwocomponent(complex)analysisofRef.[41].

4. Correlationfunctions

We nowderive expressions forthe two-point andthree-point correlation functions ofthe time-delay ﬁeld. To do so, it is con-venienttointroducearescaledversion ofthecorrelationfunction

(

x

)

deﬁnedthrough

φ

i

(

x

)φ

j

(

0

)

= (

x

)φ

2rms

δ

i j

.

(28)

By deﬁnition

(

0

)

=

1,and aswe vary x,

covers the interval

(

x

)

∈ (

0

,

1

]

.(Ref.[41]usedadifferentconventionwhere

covers theinterval

(

x

)

∈ (

0

,

1

/

2

]

.)

In

Fig. 3

,we showaplotof

attend asafunctionofx,

mea-suredinHubblelengths(H−1),forvariouscombinationsofmasses.

4.1. Two-pointfunction

We now express the time-delay correlation functions as a powerseriesin

and1

/N

.Analternativederivationofthepower spectra of the time-delay ﬁeld in terms of an integral, which is closertothelanguageofRef.[41],canbefoundin

Appendix C

.

Usingtheaboveapproximationfor

δ

t ineq.(27),thetwo-point correlationfunctionforthetime-delayis

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

=

ln

φ

x

· φ

x

N

φ

rms2

ln

φ

0

· φ

0

N

φ

2rms

,

(29)

wherewehaveusedtheabbreviatednotation

φ

x

≡ φ(

x

)

.The

two-point function will include a constant (independent of x) for a non-zero

δ

t

. This can be reabsorbed into a shift in t0, whose dependencewe havesuppressed here, andsowe will ignorethe constant in the following computation. This means that we will computetheconnected partofthecorrelationfunctions.

Wewouldliketoformanexpansion,butwedonothavea clas-sicaltrajectoryaboutwhichtoexpand.Insteadweusethe follow-ing idea:werecognizethat

φ · φ

shouldbecentralizedaroundits meanvalueof

N φ

2

rms,plusrelativelysmallﬂuctuationsatlarge

N

.

Thismeansthatitisconvenienttowrite

φ · φ

N

φ

2rms

=

1

+

φ · φ

N

φ

2rms

−

1

(30) andtreattheterminparenthesisontherightassmall,asit rep-resents the ﬂuctuationsfrom the mean. Thisallows usto Taylor expandthelogarithminpowersof

≡

φ · φ

N

φ

2rms

−

1

,

(31)

with

=

0.Nowrecallthattheseriesexpansionofthelogarithm forsmall

isln

(

1

+)

=

−

₂2

+

₃3

−. . .

,allowingustoexpand eq.(29)toanydesiredorderin

.

Theleadingnon-zeroorderisquadratic

∼

2_._It_is

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

2

= (

x

)(

0

)

,

=

φ

i

(

x

)φ

i

(

x

)φ

j

(

0

)φ

j

(

0

)

(

N

φ

rms2

)

2

−

1

,

(32)

wherewe are implicitlysumming overindicesinthesecond line (for simplicity, we will place all component indices

(

i

,

j

)

down-stairs). Using Wick’s theorem to perform the four-point contrac-tion,weﬁndtheresult

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

2

=

2

(

x

)

N

,

(33)

wherex

≡ |

x

|

.Thisprovidestheleadingapproximationforlarge

N

, or small

. Thisshould be contrasted to the RSG approximation usedinRefs.[21,22]andsummarizedin[41],inwhichthe correla-tionfunctionisapproximatedas

∼

,ratherthan

∼

2_,_at_leading order.

Forbrevity, we shallnot go throughthe resultateach higher orderhere,butwereportonresultsathigherorderin

Appendix A

. Bysummingthoseresultstodifferentorders,weﬁnd

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

=

2

N

+

2

4

N

2

−

4

N

3

+

16

6 3

N

3

+

8

4

N

4

−

32

6

N

4

+

24

8

N

4

+

O

1

N

5

(34) (where

= (

x

)

here). We ﬁnd that various cancellations have occurred. For example, the

−

8

2

/N

2 termthat enters at cubic order(see

Appendix A

)hascanceled.

Note that at a given order in 1

/

N

there are various powers of

2.However,bycollectingalltermsatagivenpowerin

2,we can identifya patterninthe valueof itscoeﬃcientsasfunctions of

N

.Weﬁnd

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

=

2

(

x

)

N

+

2

4

(

x

)

N

(

N

+

2

)

+

16

6

(

x

)

3

N

(

N

+

2

)(

N

+

4

)

+

_N

24

8

(

x

)

(

N

+

2

)(

N

+

4

)(

N

+

6

)

+ . . .

(35) Wethenidentifytheentireseriesas

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

=

∞

n=1

(7)

wherethecoeﬃcientsare Cn

(

N

)

=

1 n2

_N 2

−

1

+

n n

−1

,

(37)

with

a_b

thebinomialcoeﬃcient.Thisseriesisconvergentforany

N

and

∈ (

0

,

1

]

,andisoneofourcentralresults.

Forthecaseofasingle scalaroradoublet(complexﬁeld),the fullseriesorganizesitselfintoknownfunctions.For

N =

1 weﬁnd

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

=

2

(

x

)

+

2

4

₍

_x

₎

3

+

16

6

(

x

)

45

+ . . .

=

2

(

sin−1

(

x

))

2

.

(38) For

N =

2 weﬁnd

(

2

λ)

2

δ

t

(

x

)δ

t

(

0

)

=

2

(

x

)

+

4

₍

_x

₎

22

+

6

(

x

)

32

+ . . .

=

Li2

(

2

(

x

)),

(39)

whereLis

(

z

)

is the polylogarithmfunction. We also ﬁndthat for

anyevenvalueof

N

,theseriesisgivenbythepolylogarithm func-tionplusapolynomialin

;thisisdescribedin

Appendix B

.

Usingthepowerseries,wecaneasily obtainplotsofthe two-pointfunction forany

N

. Forconvenience,we plotthe re-scaled quantity

N (

2

λ)

2

_δ

_t

₍

_x

_)δ

_t

₍

₀

₎

_as_a_function_of

_in

_{Fig. 4}

_(top) _for different

N

.We seeconvergenceofallcurvesaswe increase

N

, whichconﬁrms thattheleading behavior ofthe(un-scaled) two-pointfunctionis

∼

1

/N

.In

Fig. 4

(bottom)weplot

N δ

t

(

x

)δ

t

(

0

)

asafunctionofx fordifferentmassesandtwodifferent

N

.

4.2.Three-pointfunction

The three-point function is givenby a simple modiﬁcation of eq.(29),namely

−(

2

λ)

3

δ

t

(

x

)δ

t

(

y

)δ

t

(

0

)

=

ln

φ

x

· φ

x

N

φ

rms2

ln

φ

y

· φ

y

N

φ

rms2

ln

φ

0

· φ

0

N

φ

rms2

.

(40)

Herewe will work only toleading non-zero order, whichin this caseiscubic.Weexpandthelogarithmsasbeforetoobtain

−(

2

λ)

3

δ

t

(

x

)δ

t

(

y

)δ

t

(

0

)

3

= (

x

)(

y

)(

0

)

,

=

φ

i

(

x

)φ

i

(

x

)φ

j

(

y

)φ

j

(

y

)φ

k

(

0

)φ

k

(

0

)

(

N

φ

2rms

)

3

+

2

−

_φ

i

(

x

)φ

i

(

x

)φ

j

(

y

)φ

j

(

y

)

(

N

φ

rms2

)

2

+

2 perms

,

(41)

where “perms” is short for permutations under interchanging x, y, 0. Using Wick’s theorem to perform the various contractions, weﬁndtheresult

−(

2

λ)

3

δ

t

(

x

)δ

t

(

y

)δ

t

(

0

)

3

=

8

(

|

x

−

y

|) (

x

) (

y

)

N

2

.

(42)

Wewouldnowlike tousethethree-point functionasa mea-sureofnon-Gaussianity.Forasinglerandomvariable,ameasureof non-Gaussianityistocomputea dimensionlessratioofthe skew-ness to the 3/2 power of the variance. For a ﬁeld theory, we symmetrize over variables, and deﬁne the following measure of non-Gaussianityinpositionspace

S

≡

√

δ

t

(

x

)δ

t

(

y

)δ

t

(

0

)

δ

t

(

x

)δ

t

(

y

)

δ

t

(

x

)δ

t

(

0

)

δ

t

(

y

)δ

t

(

0

)

.

(43)

Fig. 4. Top: a plot of the (re-scaled) two-point function of the time-delay

N (2λ)2_δ_t₍_x_)δ_t₍₀₎_as_a_function_of_{∈ [}₀_,₁_]_as_we_vary_N_:_dot-dashed_is_N₌_1, solidisN=2,dottedisN=6,anddashedisN→ ∞.Bottom:aplotofthe (re-scaled)two-pointfunctionofthetime-delayNδt(x)δt(0)asafunctionofx for differentmasses:blueism0=2 andmψ=1/2,redism0=4 andmψ=1/2,green ism0=2 andmψ=1/4,orangeism0=4 andmψ=1/4,withsolidforN=2 anddashedforN→ ∞.(Forinterpretationofthereferencestocolorinthisﬁgure legend,thereaderisreferredtothewebversionofthisarticle.)

(Recall that we are ignoring

δ

t

, as it can be just re-absorbed into t0, so the three-point and two-point functions are the con-nectedpieces.) Computingthisattheleadingorderapproximation usingeqs.(33),

(42)

(validforlarge

N

,orsmall

),weﬁnd S

≈ −

8

N

.

(44)

Curiously, thedependenceon x,y has dropped outat thisorder. Weseethatforsmall

N

thereissigniﬁcantnon-Gaussianity,while for large

N

the theory becomes Gaussian, asexpected from the centrallimittheorem.

4.3. Momentumspace

Letusnowpresentourresultsink-space.Weshallcontinueto analyzetheresultsathigh

N

,orsmall

,whichallowsustojust includetheleadingorderresults.

(8)

Fig. 5. ThedimensionlesspowerspectrumN Pδt(k)atlargeN asafunctionofk

(inunitsofH )fordifferentchoicesofmasses:blueism0=2 andmψ=1/2,redis m0=4 andmψ=1/2,ism0=2 andmψ=1/4,orangeism0=4 andmψ=1/4. (Forinterpretationofthereferences tocolorinthisﬁgurelegend,thereader is referredtothewebversionofthisarticle.)

For the two-point function, we deﬁne the power spectrum through

δ

t

(

k1

)δ

t

(

k2

)

= (

2

π

)

3

δ(

k1

+

k2

)

Pδt

(

k1

).

(45)

We useeq.(33) andFouriertransformtok-space usingthe con-volutiontheorem.Todosoitisconvenienttointroducea dimen-sionlessﬁeld

˜φ

i

≡ φ

i

/φ

rmsandintroducethecorrespondingpower

spectrum P_˜φ

(

k

)

=

Pφ

(

k

)/φ

rms2

= |

uk

|

2

/φ

2rms, which is the Fourier

transformof

(

x

)

.Wethenﬁndtheresult

Pδt

(

k

)

≈

1 2

λ

2

_N

_d3_k

(

2

π

)

3P¯φ

(

k

)

P¯φ

(

|

k

−

k

|).

(46) Adimensionlessmeasureofﬂuctuationsisthefollowing

P

δt

(

k

)

≡

k3H2Pδt

(

k

)

2

π

2

.

(47)

Thefactorofk3

_/(

_2π2

₎

_is_appropriate_as_this_gives_the_variance_per loginterval:

(

H

δ

t

)

2

=

dln k

P

δt

(

k

)

.Bystudyingeq.(46),onecan

show Pδt

≈

const forsmallk andfallsoffforlargek. Thiscreates

a spikein

P

δt

(

k

)

ata ﬁnitek∗ andits amplitude is ratherlarge.

(This is to be contrasted with the usual ﬂuctuationsin de Sitter space, Pδt

(

k

)

∝

1

/

k3,making

P

δt

(

k

)

approximatelyscaleinvariant.)

Weseethattheamplitudeofthespikescalesas

∼

1

/N

,andsoit isreducedforlarge

N

.Aplotof

P

δt

(

k

)

isgivenin

Fig. 5

.

Forthethree-pointfunction,wedeﬁnethebispectrumthrough

δ

t

(

k1

)δ

t

(

k2

)δ

t

(

k3

)

= (

2

π

)

3

δ(

k1

+

k2

+

k3

)

Bδt

(

k1

,

k2

,

k3

)

(48) wherewehaveindicatedthatthebispectrumonlydependsonthe magnitudeofthe3k-vectors,withtheconstraintthatthevectors sum to zero. We use eq. (42) and Fouriertransform to k-space,

againusingtheconvolutiontheorem.Weﬁnd Bδt

(

k1

,

k2

,

k3

)

≈ −

1 3

λ

3

_N

2

×

_d3_k

(

2

π

)

3P¯φ

(

k

)

P¯φ

(

|

k1

−

k

|)

P¯φ

(

|

k2

+

k

|) +

2 perms

(49)

Fig. 6. Thedimensionlessbispectrum√N FNLasafunctionofk (inunitsofH )at

largeNform0=4 andmψ=1/2.

Tomeasurenon-Gaussianityink-space,itisconventionalto in-troducethedimensionless fNLparameter,deﬁnedas1

fNL

(

k1

,

k2

,

k3

)

≡

Bδt

(

k1

,

k2

,

k3

)

Pδt

(

k1

)

Pδt

(

k2

)

+

2 perms

.

(50)

BysubstitutingtheaboveexpressionsforPδt andBδt,weseethat

fNLisindependentof

N

atthisleadingorder.However,thisbelies

the truedependenceofnon-Gaussianity onthe numberofﬁelds. Thisisbecause fNLisaquantitythatcanbelargeevenifthe

non-Gaussianity is relatively small (for example, on CMB scales, any

fNL smaller than

O(

105

)

isa smalllevel ofnon-Gaussianity).

In-stead a more appropriate measureof non-Gaussianity in k-space

is to compute some ratio ofthe bispectrumto the 3/2power of thepowerspectrum,analogoustothepositionspacedeﬁnitionin eq.(43).Forthesimpleequilateralcase,k1

=

k2

=

k3,wedeﬁne

F

NL

(

k

)

≡

k3/2Bδt

(

k

)

3

√

2

π

Pδt

(

k

)

3/2

,

(51)

where we have inserted a factorof k3/2

_/(

√

₂

_π

₎

_from _measuring theﬂuctuationsper loginterval.Usingeqs.(46),

(49)

,weseethat

F

NL

∝

1

/

√

N

,aswefoundinpositionspace.In

Fig. 6

,weplotthis function.Wenotethatalthoughthenon-Gaussianitycanbelarge, the peakis ona lengthscale that issmallcomparedto theCMB andsoitevadesrecentbounds

[6]

.

5. Constraintsonhybridmodels

Hybridinﬂationmodelsmustsatisfyseveralobservational con-straints.Herewediscusstheseconstraints,includingtherolethat

N

plays,anddiscusstheimplications forthescaleofthedensity spike.

5.1. Topologicaldefects

Theﬁrstconstraintonhybridmodelsconcernsthepossible for-mation of topological defects. Since the waterfall ﬁeld starts at

φ

=

0 and then falls to some vacuum, it spontaneously breaks

1 _A_factor_of_6/5_is_often_included_when_studying_the_gauge_invariant_quantity

ζ

(9)

a symmetry. For a single ﬁeld

N =

1, this breaks a

Z

2 symme-try; see Fig. 1.For multiple ﬁelds

N >

1, this breaks an SO

(

N )

symmetry.

N =

1 leads to the formation of domainwalls, which areclearlyruledout observationally,sothesemodelsarestrongly disfavored;

N =

2 leadstothe formationofcosmicstrings,which have not been observed and if they exist are constrained to be smallinnumber.Thesubjectofcosmicstringproductioninhybrid inﬂationandtheirsubsequenteffectsisdiscussedin[23].For cur-rentboundsoncosmologicaldefectsfromthePlanckcollaboration the readercan refer to [48] and the references therein.Diluting cosmicstrings tomakethem unobservableinourcasewould re-quireavery large numberofe-foldings ofthewaterfallphase to makecompatiblewithobservations,andseemsunrealistic.Further increasingthenumberofwaterfallﬁelds,

N =

3 leadstothe for-mationofmonopoles,whicharesomewhatlessconstrained;

N =

4 leads to theformation of textures,which are relatively harmless;

N >

4 avoids topological defects altogether. So choosing several waterfallﬁelds is preferred by currentconstraints on topological defects.

Evenfor

N >

4,onemightbeconcernedaboutconstraintsfrom there-orderingof

N −

1 Goldstonemodesarisingfromthe break-ing of the SO

(

N )

global symmetry. However it is important to note that all globalsymmetries are only ever approximate.So it is expected that these modes are not strictly massless, but pick up a small mass at some order, as all Goldstones do. The only importantpoint is that the mass ofthe Goldstones mG is small

comparedtothemassscalesoftheinflatonandthewaterfallfield. Forexample, ifwe have a scale of inflation with H

∼

1012GeV, thenthe inﬂatonandwaterfall ﬁeldsshould havea massnot far from

∼

1012GeV also. We then only needthe Goldstonemasses tosatisfymG

1012GeV forouranalysistobetrue,andforthe

SO

(

N )

symmetry to be agood approximationfor thepurposeof thephasetransition.Atlatetimes,these(approximate)Goldstones willappearheavy andrelaxto thebottomoftheir potential, and be harmless.So,withthis inmind, alarge VEVfor thewaterfall ﬁeldisindeedobservationallyallowed,andlargee-foldsafterthe transitionleadingtolargeblackholesisindeedallowed.

5.2.Inﬂationaryperturbations

Inflationgeneratesfluctuationsonlargescaleswhicharebeing increasingly constrainedby data.An importantconstraint onany inflation modelisthe boundon thetensor-to-scalarratio r.CMB measurementsfromPlanckplaceanupperboundontensormodes ofr

<

0

.

11 (95%confidence)[6].Theamplitudeoftensormodesis directlysetby theenergydensityduring inflation.Typical hybrid modelsareatrelativelylowenergyscales,withouttheneedfor ex-tremefinetuning,andsotheyimmediatelysatisfythisbound. Re-centdatabytheBICEP2experiment[42]claimadetectionof grav-itationalwaveswithr

∼

0

.

2 atahighconﬁdencelevel.Goingback tothesimplestpotentialofhybridinﬂation V

(ψ)

=

V0

+

1₂m2_ψ

ψ

.2

..

,

wehave sofar considered theregime V0

1₂m2ψ

ψ

2,which

gen-erates a very small amount of tensor modes. However one can operateinaregimewithV0

12m2ψ

ψ

2,wherethetensortoscalar

ratiocanbepushedtobecloserto

O(

0

.

1

)

.Furthermore,new real-izationsofhybridinﬂation[49,50]cangenerateappreciabletensor modeswithappropriate parameter choices. Since subtracting the dustforegroundhasreducedthedetectedsignaltoanupdated up-perboundonthetensormodes[43,44],we willleaveacomplete discussionof producing thecorrect value ofr in hybrid inﬂation modelsforfuturework.

Althoughthedetectionoftensormodesisnotconﬁrmed,scalar modes are pinned down with great accuracy. The tilt of the scalarmode spectrum ischaracterized by theprimordial spectral

index ns.WMAP[5]andPlanckmeasurements[6]placethe

spec-tralindexnear

ns,obs

≈

0

.

96

,

(52)

givingared spectrum.Here weexamine theconstraintsimposed onhybridmodelsinordertoobtainthisvalueofns.

Thetiltonlargescalesisdeterminedbythetimerﬁeld

ψ

.For lowscalemodelsofinﬂation,such ashybridinﬂation,the predic-tionforthespectralindexis

ns

=

1

+

2

η

,

(53) where

η

≈

1 8

π

G V_ψ V0

=

V_ψ 3 H2

.

(54)

This is to be evaluated Ne e-foldings from the end of

inﬂa-tion, where Ne

=

50–60 in typical models. Combining the above

equations, we need to satisfy V_ψ

≈ −

0

.

06H2. If we take Vψ

=

m2

ψ

2

/

2,thenVψ

>

0,andns

>

1,whichisruledout.Soweneed

higherordertermsinthepotential tocauseittobecomeconcave downatlargevaluesof

ψ

where

η

isevaluated,whileleavingthe quadratic approximation for Vψ valid atsmall

ψ

. For most

rea-sonablepotentialfunctions,suchaspotentialsthatﬂattenatlarge ﬁeldvalues,weexpect

|

V_ψ

|

m2_ψ.Sothissuggestsabound

m2_ψ

0

.

06 H2

,

(55)

whichcanonlybeavoidedbysignificantfinetuningofthe poten-tial.Hencealthoughthe timerfieldisassumedlight(mψ

<

H ), it

cannotbeextremelylight.

Asan exampleletusconsiderapotentialthatcan be approxi-matedby V

(ψ)

=

1₂m2_ψ

ψ

2

−

g₄2

ψ

4.Weareneglectinghigherorder terms needed to stabilize the potential. Let ustake V

(ψ

CMB

)

=

m2

ψ

−

3g2

ψ

CMB2

<

0, where

φ

CMB is the field value at the point where the CMB fluctuationsexit the horizon during inflation, in order to produce a red-tilted spectrum. This requirement leads to m2_ψ

3g2

ψ

_CMB2 which inturns implies V

(ψ

CMB

)

∼

m2ψ. Since

ﬂuctuations that are imprinted on the CMB must exit the hori-zon suﬃciently before the waterfall transition, in order for the CMBspectrumtokeepitsapproximatescale-invariance,the quar-tictermin thepotential considered inthisexample willbe sub-dominant during the waterfall transition, since

ψ

waterfall

< ψ

CMB. Anexamplepotentialforthetimerﬁeldis

Vψ

(ψ )

=

F2m2_ψ

2 ln

(

1

+ ψ

2

_/

_F2

_).

₍₅₆₎

IfoneTaylorexpandsthis,oneﬁndstheneededm2

ψ

2

/

2 quadratic

piece,plus the higherorder correctionsthat renderthepotential

concavedown, which is compatible with the observed red-tilted spectrum.

Sinceadetailedcalculationoftheprimordialblackhole produc-tionforspeciﬁcrealizationsofourmodel(alongwithconnectingit topotentialobservableslikesupermassiveblackholes)isdeferred forafuturepresentation,thecorrespondingchoiceofpotential pa-rameterswillbedoneatthattimeaswell.

5.3. Implicationsforscaleofspike

The length scale associatedwiththe spike inthe spectrumis setby theHubblelengthduring inﬂation H−1_,_red-shifted_by _the numberofe-foldingsofthewaterfallphase Nw.SincetheHubble

scale during inﬂation istypically microscopic,we need the dura-tionofthewaterfallphase Nw tobesigniﬁcant(e.g.,Nw

∼

30–40)

(10)

toobtainaspikeonastrophysicallylargescales.Hereweexamine ifthisispossible.

Sincewe havedeﬁnedt

=

0 tobe whenthe transitionoccurs (

ψ

= ψ

c),then Nw

=

Ht withﬁnalvalue att

=

tend.Todetermine

theﬁnalvalue,wenotethatmodesgrowattherate

λ

,derived ear-lierineq.(22).Formψ

<

H ,wecanapproximatetheparameter p

(eq.(19))asp

≈

m2_ψ

/

3H .Usingtheabovespectralindexboundin eq.(55),weseethatforsigniﬁcantlylarge Nw (e.g., Nw

∼

30–40)

theexponentialfactor exp

(

−

2 p t

)

≈

exp

−

2 m 2 ψNw 3 H2

(57) issomewhatsmallandwewillignore ithere.Inthisregime,the dimensionlessgrowthrate

λ/

H canbeapproximatedasaconstant

λ

H

≈ −

3 2

+

9 4

+

m2 0 H2

.

(58)

The typical starting value for

φ

is roughly of order H (de Sitter ﬂuctuations) andthe typical end value for

φ

is roughly of order

MPl(Planckscale).Forselfconsistency,

φ

mustpassfromits

start-ingvalue toitsendvalue inNw e-foldingswithratesetby

λ/

H .

ThisgivestheapproximatevalueforNw as2 Nw

≈

H

λ

ln

MPl H

.

(59)

Thishasaclearconsequence:Ifwechoosem0

H ,asisdone in somemodelsofhybridinﬂation,then H

/λ

1.Sounlesswepush

H tobe many,manyordersofmagnitudesmallerthen MPl,then

Nw willberathersmall.Thiswillleadtoaspikeinthespectrum

onrathersmallscalesandpossiblyignorabletoastrophysics. Notethatifwehadignoredthespectralindexboundthatleads toeq. (55),then wecould havetakenmψ arbitrarily small,

lead-ing toan arbitrarily small p value.In this(unrealistic)limit,itis simpletoshow

λ

H

≈

2 m2 ψm20Nw 9 H4

.

(60)

Sobytakingmψ arbitrarilysmall,

λ

couldbemadesmall,andNw

couldeasilybemadelarge.However,theexistence ofthespectral indexboundessentiallyforbidsthis,requiringustogoina differ-entdirection.

The only way to increase Nw and satisfy the spectral index

boundonmψ istotakem0somewhatclosetoH .ThisallowsH

/λ

tobeappreciablefromeq.(58).Forinstance,ifwesetm0

=

1

.

3H , then H

/λ

≈

2. Ifwethen take H justafew orders ofmagnitude below MPl, say H

∼

10−6,7MPl, which is reasonable for inﬂation

models,we canachieve a signiﬁcantvalue for Nw.Thiswilllead

to aspike in thespectrum on astrophysically largescales, which ispotentiallyquiteinteresting.Itispossiblethattherewillbe dis-tortionsinthespectrumbytakingm0 closeto H ,butwewillnot explorethose detailshere. However,there isanimportant conse-quencethatweexploreinthenextsubsection.

5.4. Eternalinﬂation

Since we are being pushed towards a somewhat low value of m0,near H ,we needto check ifthetheory still makes sense. One potential problem isthat the theory mayenter a regime of

2 _A_better_{approximation}_comes_from_tracking_the_full_time_dependence_of_λ_and integratingtheargumentoftheexponentialexp(tdt λ(t )),butthisapproximation suﬃcesforthepresentdiscussion.

eternalinflation.Thiscouldoccurforthewaterfallfieldatthe hill-top. This would wipe out information of the timer field, which establishedtheapproximatelyscaleinvariantspectrumon cosmo-logicalscales.

The boundary for eternal inﬂation is roughly when the den-sityﬂuctuationsare

O(

1

)

,andthisoccurswhentheﬂuctuationsin the time delay are

(

H

δ

t

)

2

=

O(

1

)

.Toconvertthis intoa lower bound on m0, letusimagine thatm0 is even smallerthan H .In thisregime,thegrowthrate

λ

can beestimatedusingeq. (58)as

λ

∼

m2₀

/

H2.Usingeq. (33)thisgives

(

H

δ

t

)

2

∼

H4

/(N

m4₀

)

.This impliesthateternalinﬂationoccurswhenthewaterfallmassis be-lowacriticalvaluemc₀,whichisroughly

mc₀

∼

H

N

1/4

.

(61)

Sowhen

N ∼

1 wecannothavem0 nearH ,becausewethen en-tereternalinﬂation.Ontheotherhand,forlarge

N

weareallowed to havem0 near H andavoidthisproblem. Thismakessense in-tuitively,because formanyfieldsitis statisticallyfavorable forat least one ofthe fieldsto fall offthe hill-top,causinginflation to end. Hencelarge

N

ismoreeasilycompatiblewiththeaboveset ofconstraintsthanlow

N

.

6. Discussionandconclusions

In thisworkwe studied densityperturbations inhybrid inﬂa-tion caused by

N

waterfall ﬁelds,which contains a spike in the spectrum.We derivedexpressions forcorrelationfunctionsofthe time-delayandconstrainedparameterswithobservations.

Densityperturbations:Wederivedaconvergentseriesexpansion in powers of 1

/N

and

(

x

)

, thedimensionless correlation func-tionfortheﬁeld,forthetwo-point functionofthetime-delayfor any

N

, and the leading order behavior of the three-point func-tionofthetime-delayforlarge

N

.Thesecorrelationfunctionsare wellapproximatedbytheﬁrsttermintheseriesforlarge

N

(even for

N =

2 theleading termismoderatelyaccurateto

∼

30%,and muchmoreaccurateforhigher

N

).Inthisregime,theﬂuctuations aresuppressed,withtwo-pointandthree-pointfunctionsgivenby

δ

t

(

x

) δ

t

(

0

)

≈

2

₍

_x

₎

2

λ

2

_N

,

δ

t

(

x

) δ

t

(

y

) δ

t

(

0

)

≈ −

(

|

x

−

y

|)(

x

)(

y

)

λ

3

_N

2

.

(62)

Although thisreducesthe spikein the spectrum,forany moder-atevalueof

N

,such as

N =

3

,

4

,

5,theamplitudeofthespikeis still quitelarge (ordersofmagnitudelargerthanthe

∼

10−5 level ﬂuctuationson larger scales relevantto theCMB), andmayhave signiﬁcantastrophysicalconsequences.Also,therelativesizeofthe three-point function to the 3/2 power of the two-point function scales as

∼

1

/

√

N

.Inaccordancewiththecentral limittheorem, theﬂuctuationsbecomemoreGaussianatlarge

N

.Thiswillmake theanalysisofthesubsequentcosmologicalevolutionmore man-ageable,asthisprovides a simplespectrumforinitial conditions. We note that since we are considering small scales compared to theCMB,thenthisnon-GaussianityevadesPlanckbounds[6].

Constraints: We mentioned that hybrid models avoid topolog-ical defects for large

N

, while tensor mode constraints can be satisﬁed in differentmodels. Avery serious constrainton hybrid models comesfromthe observedspectral indexns

≈

0

.

96,which

requires the potential to flatten at large field values. One conse-quence of thisis that thetimer field mass mψ needs to be only

a little smallerthan theHubbleparameter during inflation H ,or else themodelissignificantlyfinetuned.Foralargevalue ofthe waterfall field mass m0, thiswould implya large growth rateof