Elementary Theorems Regarding Blue Isocurvature Perturbations

0.61 MB·English

by  Daniel J. H. Chung

Elementary Theorems Regarding Blue Isocurvature Perturbations
Daniel J. H. Chung1,2,∗ and Hojin Yoo1,3,4,†
1Department of Physics, University of Wisconsin, Madison, WI 53706, USA
2Kavli Institute for Cosmological Physics,
University of Chicago, Chicago IL 60637, USA
5 3Berkeley Center for Theoretical Physics,
1
0 University of California, Berkeley, CA 94720
2
n 4Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
a
J
BlueCDM-photonisocurvatureperturbationsareattractiveintermsofobservabilityand
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may be typical from the perspective of generic mass relations in supergravity. We present

]
O andapplythreetheoremsusefulforblueisocurvatureperturbationsarisingfromlinearspec-
C
tatorscalarfields. Intheprocess,wegiveamorepreciseformulaforthebluespectrumasso-
.
h
p ciatedwiththeaxionmodelof0904.3800,whichcaninaparametriccornergiveafactorof
-
o
O(10) correction. We explain how a conserved current associated with Peccei-Quinn sym-
r
t
s
a metry plays a crucial role and explicitly plot several example spectra including the breaks
[

in the spectra. We also resolve a little puzzle arising from a naive multiplication of isocur-
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v
vatureexpressionthatshedslightonthegravitationalimprintoftheadiabaticperturbations
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6 onthefieldsresponsibleforblueisocurvaturefluctuations.
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.
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1 Contents
:
v
i
X 1. Introduction 2
r
a
2. UsefulSimpleTheoremsforBlueIsocurvatureModelswithaSlowlyRollingTime
DependentVEV 5
2.1. Definitions 5
2.2. Theorem1: ClassicallyConservedIsocurvatureQuantity 8
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
2.3. Theorem2: GaugeInvariantIsocurvatureSpectrumDuringRadiationDomination 13
2.4. Theorem3: QuantumCorrelatorofLinearSpectatorIsocurvaturePerturbations 14
3. Applications 27
3.1. ImprovementoftheAxionBlueIsocurvatureScenario[1] 27
3.1.1. AReviewoftheAxionBlueIsocurvatureScenario[1] 28
3.1.2. Improvement 30
3.2. DoDressingEffectsGiveaLowerBoundontheBlueIsocurvatureSpectrum? 46
4. SummaryandConclusion 48
Acknowledgments 49
A. ParticularSolutionIntheSubhorizonRegion 49
B. Mixture 51
C. Slow-Rollof χ 52
References 53
1. INTRODUCTION
Single scalar field inflationary models generate approximately adiabatic, scale-invariant, and
Gaussian primordial density perturbations [2–10]. This is consistent with the Cosmic Microwave
Background (CMB) measurements [11–22] and the Large Scale Structure (LSS) observations
[23, 24]. However, non-thermal cold dark matter (CDM) scenarios such as axions [25–27] and
WIMPZILLAs [28–33] naturally have observable CDM-photon isocurvature perturbations (e.g.
[34–49]) since the CDM never thermalizes with the photons. Indeed, it is remarkable that a sub-
dominant dark matter component as small as 10−4 of the total dark matter content can leave an
experimentally detectable effect through cosmology (see e.g. [50]). Furthermore, isocurvature
perturbations are interesting since it can generate rich density perturbation phenomenology. For
example, unlike standard single field inflationary scenarios, degrees of freedom responsible for
isocurvatureperturbationsareabletogeneratelargeprimordiallocalnon-Gaussianities[50–78].
3
Scale-invariant isocurvature spectrum is well constrained as its power on CMB length scales
has to be less than about 3% of the adiabatic power [13, 16, 79–84]. Because the largest scale
invariant isocurvature effects on CMB measurements occurs on long length scales (e.g. see the
appendix in [85]), one expects scale invariant isocurvature effects to be well hidden in any future
observations probing short length scales. However, if the isocurvature spectrum is very blue, then
isocurvatureeffectsthatarehiddenonlonglengthscalesmaybecomelargeeffectsonshortlength
scales (see e.g. [86–89]). If such strongly blue spectral index isocurvature signal is uncovered in
the future, one may ask what one will learn regarding the high energy physics of the isocurvature
sector.
Oneanswertothatisgivenby[1]inwhichasupersymmetricaxionmodelisconstructedgiving
rise to a blue spectrum. In that work, the phenomenologically relevant axion isocurvature pertur-
bation amplitude δa is assumed to be given by the frozen value δa/ϕ at horizon crossing where
+
ϕ is the classical value of the radial field that breaks the Peccei-Quinn (PQ) symmetry during
+
inflation. Unlike in the conventional minimal axion scenario in which the order parameter ϕ is
+
sitting at its potential minimum during inflation, its value is initially displaced from the minimum
andisslowlydecreasingtowardsthestableminimumduringinflation. Hence,theassumedfrozen
value at horizon crossing increases for larger wave vector k modes which leave the horizon later.
In that way, a blue spectrum is generated over a k range that depends on the spectral index which
controlstheamountoftimeϕ takestosettletoitsminimum. Furthermore,becausesupergravity
+
structure generically induces a Hubble scale mass [90] for ϕ , the spectral index can be easily
+
extremelyblue. Forexample,ithasbeenclaimedthatthisscenarioallowsanisocurvaturespectral
indexofn=4fork∈[kmin,kmax ]specifiedbykmax/kmin ∼exp(10).
Partly motivated by this result, we formulate three elementary “theorems” regarding isocurva-
tureperturbationswithaverybluespectrumfornon-thermaldarkmatterfieldssuchastheaxions
with a displaced (possibly time dependent) vacuum expectation value during inflation.1 Theorem
1 defines a superhorizon conserved quantity for systems possessing an approximate symmetry
associatedwithlinearlyperturbedsystem. Themeritofthistheoremcomparedtopreviousdiscus-
sions of this topic in the literature (see e.g. [91, 92]) is its ability to go beyond the end of inflation
and the reheating process. Theorem 2 describes under what averaging conditions that fluid quan-
tities behave as δχ /χ . This second theorem merely restates what is known in the literature
nad 0
1Theproofsareonlyattherigorofatypicalphysicsliterature.
4
in the context of the theorem. Theorem 3 describes the computation of the quantum isocurvature
perturbations. The merit of theorem 3 compared to the previous discussion in the literature is the
explicitcanonicalquantizationinthepresenceoflinearizedgravitationalconstraints. Furthermore,
wepointouttheclearconditionsunderwhichthesimpleanalyticestimatesarevalid.
We then give couple of applications of our theorems. First, we improve on the naive quanti-
zation of axions in scenario of [1] and compute O(1+ 1 ) corrections to the spectrum of with
(n−4)2
spectral index n. In the process, an interesting application of conserved PQ symmetry current is
made,whichexplainshowtwoindependentdynamicaldegreesoffreedombehaveasasingleone
duringafinitetimedurationofinterest. Thetheoremsalsohelptosetapreciseboundaryofwhere
the simple analytic computations are invalid. For example, contrary to claims of [1], n=4 spec-
trum cannot be generated in their scenario. Another consequence of understanding the boundary
is that if the ratio of axion isocurvature blue power amplitude to the adiabatic power amplitude
is at most of the order of a few percent on the largest observable scales and can be described in
terms of quantization methods presented in this paper (and implicitly approximated in [1]), most
of the cold dark matter must be made of different species. We also illustrate through example
plots, phenomenologically interesting parametric corners of the model (having a six dimensional
parameter space). Although observable spectra can contain breaks, these break regions typically
contain k-space domains for which the simple analytic computation is invalid. We identify how
large the expansion rate H during inflation can be in this class of models generating a large blue
spectrum. A measurement of tensor-to-scalar ratio at the level of r =O(10−1) will disfavor this
classofmodels,atleastinitssimplestform.
In another application, theorem 3 is used to explain why the isocurvature blue spectrum does
not have a simple lower bound suggested by a naive operator product analysis. More explicitly,
the isocurvature perturbations are defined to be a contrast of the form S ∼C δχ−C δφ where
χ 1 2
C are background field dependent coefficients and δφ is the inflaton field and δχ is the field
i
responsible for the existence of isocurvature perturbations. In other words, the isocurvature field
isalwaysdressedwiththeinflatonsector. Thequantumcorrelator(cid:104)SS(cid:105)wouldthennaivelyhavea
piece that is proportional toC2(cid:104)δφδφ(cid:105) coming from the dressing. This piece for a blue spectrum
2
is of order of the adiabatic perturbation power spectrum. If the cross correlation piece does not
preciselycancelthispiece,(cid:104)S S (cid:105)wouldbeoftheorderofadiabaticspectrum,leadingtoasimple
χ χ
lower bound. However, the theorem shows that the power spectrum of S generically behaves
χ
independentlyoftheadiabaticspectrum. Themorebroadlessonencapsulatedbytheorem3isthat
5
the gravitational coupling of δχ to δφ makes δχ grow an inhomogeneity that looks like δφ such
thatS becomesindependentofδφ.
χ
The order of the presentation will be as follows. In Sec. 2, three simple theorems and couple
of corollaries useful for spectator dark matter isocurvature spectra are presented. In Sec. 3, a
couple of applications of the theorems are given. One application corresponds to improving and
elucidating the computation of [1]. The second application corresponds to understanding how
dressing effects coming from the definition of the isocurvature perturbations do not mix inflaton
field quantum fluctuations with the dark matter field quantum fluctuations because of the secular
growth imprinting an adiabatic inhomogeneity to the dark matter field. We close with a summary
and thoughts on future work to be done in this direction. In the appendix, we collect some results
usefulforthetheorems.
2. USEFULSIMPLETHEOREMSFORBLUEISOCURVATUREMODELSWITHASLOWLY
ROLLINGTIMEDEPENDENTVEV
2.1. Definitions
Inthissubsection,wedefinethelanguageusedforourtheorems.
Metric and Fourier Conventions Although theorems that we present are gauge invariant, we
will have the occasion to use several gauges in our proofs. The Newtonian gauge scalar perturba-
tionswillbeparameterizedas
ds2 =(1+2Ψ(N))dt2−a2(t)(1+2Φ(N))|d(cid:126)x|2. (1)
Weconsiderslow-rollinflatonfieldϕ scenariosinwhichsuperhorizonadiabaticperturbationsare
approximately conserved. Conserved adiabatic curvature perturbations on superhorizon scales is
giveninNewtoniangaugebythesolution[10,93–95]
δρ(N)(t,(cid:126)k)
Φ(N)(t,(cid:126)k)−H =ζ ≡constant (2)
ρ˙(t) (cid:126)k
whereH ≡a˙/aand|(cid:126)k/a|(cid:28)H andwehaveintroducedtheFourierconvention
ˆ
Q(t,(cid:126)k)= d3xe−i(cid:126)k·(cid:126)xQ(t,(cid:126)x). (3)
IntheNewtoniangauge,theexpansionismanifestlyisotropic. Furthermore,Φ(N) hastheintuitive
interpretation of being the gravitational potential in the Poisson equation. Because of these prop-
6
erties,thefieldequationsintheNewtoniangaugeareconvenienttoworkwithwhenworkingwith
classicalequations.
On the contrary, the spatially flat gauge is more useful for quantization during inflation (see
e.g.[96]). Thescalarmetricperturbationconventioninspatiallyflatgaugecanbechosentobe
ds2 =(1+2Ψ(sf))dt2+a∂F(sf)dtdxi−a2(t)|d(cid:126)x|2. (4)
i
As shown in Sec. 2.4, the relevant interaction action derived from solving the gravitational con-
straintsinthisgaugeconsistonlyoflocaltermsofthefieldsunlikeforthecorrespondingequations
in the Newtonian gauge. Thus, the quantization of fields and the investigation of the subhorizon
mode functions are technically simpler in this gauge. Hence, we will employ the spatially flat
gauge only for quantization during inflation which establishes the initial conditions for the late
timeclassicalequations.
Linear Spectator Isocurvature Field Let linear spectator isocurvature field be defined as a
canonicallynormalizedscalarfield χ =χ (t)+δχ(N)(t,(cid:126)x)forwhich
0
δρ(N) ∝δχ(N)+O(δχ(N)2) (5)
χ
H
(N)
χ (t )(cid:29) | (6)
0 duringinflation 2π duringinflation
δρ(N) δT(N)0
χ χ 0
= (cid:28)1 (7)
(N) (N)0
δρ δT
dominant dominant 0
inNewtoniangaugewherethesubscript“dominant”correspondstotheenergydensitycomponent
(N)0
thatdominatesT . Forexample,duringinflation,“dominant”correspondstothelabelϕ while
0
during radiation domination, “dominant” corresponds to the label γ representing the relativistic
degreesoffreedom. Becausewewillfocuson
1
V = m2χ2, (8)
χ
2
Eq.(7)translatesto
m2 χ √
0
(cid:28)3 2ε (9)
H2M
p
whereε istheinflationaryslow-rollparameter. IftheinflatonpotentialisgivenasV (ϕ),then
ϕ
M2 (cid:18)V(cid:48)(ϕ )(cid:19)2
p ϕ 0
ε = (10)
2 V (ϕ )
ϕ 0
where ϕ =ϕ (t)+δϕ. The effective expansion parameters are χ /M and slow-roll parameters
0 0 p
oftheinflatonfieldifm/H ∼O(1).
7
SpectralConventions Thegaugeinvariantspectrumoflinearspectatorχ-photonisocurvature
perturbationsusefulforBoltzmannequationsisoftendefinedduringradiationdominateduniverse
through
ˆ
k3 d3k(cid:48)
∆2 (k)≡ (cid:104)δ (t,(cid:126)k)δ (t,(cid:126)k(cid:48))(cid:105) (11)
sχ 2π2 (2π)3 sχ sχ
δ ≡3(ζ −ζ ) (12)
sχ χ γ
ζ =Φ(N)+δρχ(N)|backgroundsmoothed, ζ =Φ(N)+ δργ(N) (13)
χ γ
3(cid:104)ρ +P (cid:105) 3(ρ +P )
χ χ time γ γ
where P are pressure quantities corresponding to −Ti components of the energy momentum ten-
i i
sor. Here, the “time” average in the denominator of the definition of ζ corresponds to a time
χ
averageoverm−1 timescale. The“backgroundsmoothed”inthenumeratorofthedefinitionofζ
χ
correspondsto averagingover m−1 timescale allquadratic termsin thebackground χ (t)appear-
0
ing in the numerator. The variables ζ and ζ are conserved outside the horizon if the pressure of
χ γ
the constituent is a function only of its energy density. In particular, ζ corresponds to the gauge-
γ
invariant curvature perturbation if we assume that radiation behaves as a single component fluid
comingfromtheinflatondecay. Forsingle-fieldinflation,observationalnormalizationof
∆2(k =0.05Mpc−1) = ∆2 (k )
ζ 0 ζγ 0
V (k )
≈ ϕ 0 ≈2.4×10−9 (14)
24π2M4ε
p
corresponds to the currently known approximate value of adiabatic curvature perturbation ampli-
tude.
The χ-photon isocurvature spectrum often contain k-space domains which can be parameter-
izedas
(cid:18) (cid:19)n−1
k
∆2 (k)=∆2 (k ) (15)
sχ sχ 0 k
0
wherenisthespectralindex. Theisocurvaturespectrumisbluewhenn>1. Theprimaryfocusof
this paper is regarding spectra for which n−1(cid:38)O(0.1) which become parametrically insensitive
to the inflationary slow roll parameter values of O(ε) < 0.02. As far as the phenomenological
boundsareconcerned,notethat
∆2(k)=ω2∆2 (k) (16)
s χ sχ
where ω ≤1 is the fraction of cold dark matter that is in the χ field as is explained in Appendix
χ
B. The current phenomenological bounds on ∆2(k)/∆2 (k) for scale invariant power spectrum is
s s
χ
approximatelyafewpercent[13,16,79–84].
8
Withthesedefinitionsandassumptions,wecanconstructausefulstatementthatcanbeusedto
setclassicalequationboundaryconditionsbeforeEq.(7)breaksdown. Themostimportantofthe
three theorems that will be presented below is theorem three. Note that one of the key merits of
the theorem that we are presenting is its applicability connecting computations during inflation to
variablesduringradiationdomination.
2.2. Theorem1: ClassicallyConservedIsocurvatureQuantity
Here is a statement of the first theorem. In slow-roll inflationary scenarios, the linear spectator
isocurvaturequantity
2δχ
S (t,(cid:126)k)≡ nad (17)
χ
χ (t)
0
where
δχ ≡δχ(G)(t,(cid:126)k)−δχ(G)(t,(cid:126)k) (18)
nad ad
on superhorizon length scales is approximately conserved as long as χ interaction is dominated
by V = m2χ2/2 and gravity, anisotropic stress effects can be neglected, and attractor behavior
χ
of δχ and χ (t) is relevant during inflation (i.e. non-pathological boundary conditions are
nad 0
chosenforthehomogeneousfield)withanexpansionrateofH. Asufficientconditionforattractor
behaviorwithnon-pathologicalboundaryconditionis
|χ˙ (t )|(cid:46)m2χ (t )/H
0 iniital 0 initial
νN (cid:29)1 (19)
k
where
(cid:114)
3 4m2
ν ≡ 1− (20)
2 9H2
and N is the number of efolds between the time of k-mode horizon exit and the end of inflation.
k
Here,wehavedefined
ˆ
χ˙ (t)
δχ(G)(t,(cid:126)k)≡−ζ 0 dta(t)+ξ0∂ χ (t) (21)
ad (cid:126)k a(t) 0 0
where ξ0 = 0 in the Newtonian gauge (i.e. G = N) and in any other gauge G is related to the
Newtonian gauge coordinates through x(N)µ =x(G)µ +(ξ0,δij∂ξ). Furthermore, S is a gauge
i χ
invariant quantity. Note that we have introduced a factor of 2 in the definition of S for later
χ
9
convenience. Finally,notethatthistheoremisformulatedattheclassicalsolutionlevel. Theerror
intheconservationcomingfromtheassumptionofattractorbehaviorcanbeestimatedas
attractorerror ∼O(exp[−2νN ]) (22)
k
wherethecoefficientoftheerrordependsondetailsofinitialconditionsofboththehomogeneous
mode and the perturbation mode at the beginning of inflation. The fractional error O(E) in the
conservationisapproximatedtobethetermsthataredroppedinmakingthisstatement:
(N)
δρ
E =exp[−2νN ]+ χ . (23)
k
(N)
δρ
dominant
We also implicitly assume that the post-inflationary cosmological history consists of smoothly
connectedpatchesofpower-laws.
It is important to note that this theorem makes S conserved independently of the details not
χ
stated in the theorem, including some of the details of the end of the inflation, reheating, early
radiation domination, and how χ makes the transition from a slow-roll field to a coherently os-
0
cillating one. In particular, the classical conservation here is valid even when ε → 1 at the end
ofinflation,unlikethespatiallyflatgaugequantityδχ(sf)/χ whichundergoesgenericallyunder-
0
goestimeevolutionattheendofinflation. Theconditionsstatedinthetheoremcanbeunderstood
as a decoupling limit of the isocurvature perturbations, and this theorem establishes a classically
conservedquantityinthatlimit. Noteoneoftheimportantpointsforthispaper: thenumeratorδχ
and the denominator χ must correspond to the same dynamical degree of freedom that responds
0
to the same potential V dominated by the mass term. Finally, note that when we state the as-
χ
sumptionthatthemasstermandgravitydominatetheinteractions,wearestatingthatperturbative
interactionsaretooweaktothermalizethesystem.
proof
ConsidertheequationofmotionfortheperturbationvariableintheNewtoniangaugeδχ(N) inthe
longwavelengthlimitinwhichwecanneglectthegradientterms:
δχ¨(N)+3Hδχ˙(N)+V(cid:48)(cid:48)(χ )δχ(N)−4χ˙ Ψ˙(N)+2V(cid:48)(χ )Ψ(N) =0. (24)
χ 0 0 χ 0
Here, we have assumed that gravitational interactions and potential self-interactionsV(cid:48)(χ ) dom-
χ 0
inate the interactions. If anisotropic stress effects can be neglected, the ij component of Einstein
equationsimply
Φ(N) =−Ψ(N). (25)
10
The00componentofEinsteinequationinNewtoniangaugepartiallydeterminingΨ(N) is
a˙ 1 (cid:104) (cid:105)
−3 (HΨ(N)+Ψ˙(N))= δρ(N)+δρ(N) (26)
a 2M2 χ dominant
p
where
δρ(N) =−χ˙2Ψ+χ˙ ∂ δχ+V(cid:48)δχ (27)
χ 0 0 t χ
(N) 0(N)
andδρ ≡δT isthedominantcontributiontotheenergy-momentumtensoras
dominant dominant 0
discussedinEq.(7).2 Inthelimit
(N)
δρ
χ
(cid:28)1, (28)
(N)
δρ
dominant
weseethatΨ(N)isindependentlyofδχ. Hence,withtheconditionofEq.(28),theΨ(N)dependent
termsinEq.(24)areexternalsourcesterms. Duetodilatationdiffeomorphismgaugesolutionthat
liftstophysicalsolutions[97],thereexistsanadiabaticsolution
ˆ
χ˙ (t)
(N) 0
δχ =−ζ dta(t) (29)
ad (cid:126)k a(t)
whereζ istheusualtimeindependentgauge-invariantcurvatureperturbationconstantdetermined
(cid:126)k
by the inflaton sector ϕ approximately independently of δχ(N) as long as Eq. (28) is satisfied.
From the perspective of the classical equations we are discussing here, ζ is simply a constant
k
parameterizingasolutiontoEq.(24)wherethegravitationalpotentialisgivenbyEq.(2).3
Given that Eq. (24) is a second order differential equation, the most general solution corre-
sponds to two independent solutions h to the homogeneous equation added to the particular
1,2
solutiongivenbyEq.(29):
δχ(N) =c h(N)+c h(N)+δχ(N) (30)
1 1 2 2 ad
wherec andc arecoefficientsindependentoftime. Hence,weseethatinthelimitthatk/(aH)→
1 2
0canbeneglected,thenumeratorof
δχ δχ(N)−δχ(N) c ((cid:126)k)h(N)(t,(cid:126)k)+c ((cid:126)k)h(N)(t,(cid:126)k)
nad = ad = 1 1 2 2 (31)
χ (t) χ (t) χ (t)
0 0 0
2These statements can easily be covariantized, but such formalizations tend to obscure the intuition rather than
to illuminate the intuition. Since our aim is to illuminate the intuition of the simple physics, we will leave the
presentationintheexplicitlygaugedependentform.
3Althoughwehavenotmadeanyexplicitassumptionsaboutthebackgroundenergydensity,Eq.(28)doesdepend
onthebackgroundenergydensity.