Phenomenological constraints on axion models of dynamical dark matter

Phenomenological constraints on axion models of dynamical dark matter

Keith R. Dienes1,2,3,* and Brooks Thomas4,†

1Physics Division, National Science Foundation, Arlington, Virginia 22230, USA2Department of Physics, University of Maryland, College Park, Maryland 20742, USA

3Department of Physics, University of Arizona, Tucson, Arizona 85721, USA4Department of Physics, University of Hawaii, Honolulu, Hawaii 96822, USA

(Received 19 March 2012; published 12 September 2012)

In two recent papers [K. R. Dienes and B. Thomas, Phys. Rev. D 85, 083523 (2012)] [K. R. Dienes and

B. Thomas, Phys. Rev. D 85, 083524 (2012)] we introduced ‘‘dynamical dark matter’’ (DDM), a new

framework for dark-matter physics in which the requirement of stability is replaced by a delicate

balancing between lifetimes and cosmological abundances across a vast ensemble of individual dark-

matter components whose collective behavior transcends that normally associated with traditional dark-

matter candidates. We also presented an explicit model involving axions in large extra spacetime

dimensions, and demonstrated that this model has all of the features necessary to constitute a viable

realization of the general DDM framework. In this paper, we complete our study by performing a general

analysis of all phenomenological constraints which are relevant to this bulk-axion DDM model. Although

the analysis in this paper is primarily aimed at our specific DDM model, many of our findings have

important implications for bulk-axion theories in general. Our analysis can also serve as a prototype for

phenomenological studies of theories in which there exist large numbers of interacting and decaying

particles.

DOI: 10.1103/PhysRevD.86.055013 PACS numbers: 95.35.+d, 14.80.Rt, 14.80.Va, 98.80.Cq

I. INTRODUCTION

Dynamical dark matter (DDM) [1,2] is a new frameworkfor dark-matter physics in which the requirement of stabil-ity is replaced by a delicate balancing between lifetimesand cosmological abundances across a vast ensemble ofindividual dark-matter components. Due to the range oflifetimes and abundances of these components, their col-lective behavior transcends that normally associated withtraditional dark-matter candidates. In particular, quantitiessuch as the total dark-matter relic abundance, the propor-tional composition of the ensemble in terms of its constit-uents, and the effective equation of state for the ensemblepossess a nontrivial time dependence beyond that associ-ated with the expansion of the Universe. Indeed, from thisperspective, DDM may be viewed as the most generalpossible framework for dark-matter physics, and tradi-tional dark-matter models are merely a limiting case ofthe DDM framework in which the states which composethe dark sector are taken to be relatively few in number andtherefore stable.

In Ref. [1], we laid out the general theoretical features ofthe DDM framework. By contrast, in Ref. [2], we presentedan explicit realization of the DDM framework: a model inwhich the particles which constitute the dark-matter en-semble are the Kaluza-Klein (KK) excitations of an axion-like field propagating in the bulk of large extra spacetimedimensions. We demonstrated that this model has all of the

features necessary to constitute a viable realization of thegeneral DDM framework. In this paper, we complete ourstudy by performing a general analysis of all phenomeno-logical constraints which are relevant to this bulk-axionDDM model. Although the analysis in this paper is pri-marily aimed at our specific DDM model, many of ourfindings have important implications for theories involvinglarge extra dimensions in general. Furthermore, our analy-sis can also serve as a prototype for phenomenologicalstudies of theories in which there exist large numbers ofinteracting and decaying particles.It is important to emphasize why a general analysis of

this sort is necessary, given the existence of numerous priorstudies of the phenomenological and cosmological con-straints on axions and axionlike fields, unstable relics, andthe physical properties of miscellaneous dark-matter can-didates. As discussed in Refs. [1,2], such studies are typi-cally applicable to dark sectors involving one or only a fewfields, and are typically quoted in terms of limits on themass, decay width, or couplings of any individual suchfield. It is not at all obvious how such bounds apply to aDDM ensemble—a dark-matter candidate which is notcharacterized by a well-defined single mass, decay width,or set of couplings. For example, constraints on a cosmo-logical population of unstable particles derived from big-bang-nucleosynthesis (BBN) considerations or bounds ondistortions in the cosmic microwave background (CMB)are generally derived under the assumption that such apopulation comprises but a single particle species with awell-defined lifetime and branching fractions. Such con-straints are not directly applicable to a DDM ensemble (in

*[email protected]†[email protected]

PHYSICAL REVIEW D 86, 055013 (2012)

1550-7998=2012=86(5)=055013(44) 055013-1 � 2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevD.85.083523



which lifetimes are balanced against abundances), andmust therefore be reexamined in this new context.

In this paper, we shall develop methods for dealing withthese issues and for properly characterizing the constraintson models in which the dark-matter candidate is an en-semble of states rather than a single particle. As we shallfind, the presence of nontrivial mixings among the KKexcitations in our DDM model gives rise to a number ofsurprising effects which are ultimately critical for its phe-nomenological viability. One of these is a so-called‘‘decoherence’’ phenomenon [1–3] which helps to explainhow the dark matter in this model remains largely invisibleto detection. Another is a suppression, induced by thismixing, of the couplings between the lighter particles inthe dark-matter ensemble and the fields of the StandardModel (SM). As we shall see, these effects assert them-selves in a variety of phenomenological contexts and play acrucial role in loosening a battery of constraints whichwould otherwise prove extremely severe.

This paper is organized as follows. In Sect. II, we brieflysummarize the physics of axions in extra dimensions andreview the notational conventions established in Ref. [2],which we once again adopt in this work. In Sect. III, weexamine a number of processes, both thermal and non-thermal in nature, which contribute to the generation of acosmological population of axions. We calculate the ratesassociated with these processes and assess the relativeimportance of the associated production mechanismswithin different regions of model-parameter space. InSect. IV, we then discuss the phenomenological, astrophys-ical, and cosmological constraints relevant for bulk-axionDDM models and assess how the parameter space of ourmodel is bounded by each of these constraints. In Sect. V,

we summarize the collective consequences of these con-straints on the parameter space of our bulk-axion DDMmodel. Finally, in Sect. VI, we discuss the implications ofour results for future research.

II. GENERALIZED AXIONS IN EXTRADIMENSIONS: A REVIEW

In this section, we provide a brief review of the physicsof generalized axions in extra dimensions. (More detailedreviews can be found in Refs. [2,3].) By ‘‘generalizedaxion,’’ we mean any pseudo-Nambu-Goldstone bosonwhich receives its mass from instanton effects related toa non-Abelian gauge groupGwhich confines at some scale�G. Note that the ordinary QCD axion [4,5] is a specialcase of this, in which G is identified with SUð3Þ color and�G is identified with �QCD � 250 MeV. However, in this

paper, we shall leave these scales arbitrary in order to giveour analysis a wider range of applicability. We will alsoassume the existence of a global Abelian symmetry Uð1ÞXwhich plays the role played by the Peccei-Quinn symmetryUð1ÞPQ in the specific case of a QCD axion.

Our goal in this paper is to study the phenomenologicalconstraints that arise when a generalized axion is allowedto propagate in the bulk [3] of a theory with extra space-time dimensions [6,7]. In particular, we consider the casein which the axion propagates in a single, large, flat extradimension compactified on a S1=Z2 orbifold of radius R.The fields of the SM are assumed to be restricted to a branelocated at x5 ¼ 0. We also assume that the additional non-Abelian gauge group G is restricted to the brane at x5 ¼ 0.At temperatures T � �G, the effective action for a bulkaxion in five dimensions can be written in the form

Seff ¼Z

d4xZ 2�R

0dx5

�1

2@Ma@

Maþ �ðx5ÞðLbrane þLintÞ�; (2.1)

where ‘‘a’’ denotes our five-dimensional axion field, Lbrane contains the terms involving the brane fields alone, and Lint

contains the interaction terms coupling the brane-localized fields to the five-dimensional axion. The second of these termsis given by

L int ¼ g2G�

32�2f3=2X

aGa��

~Ga�� þXi

ci

f3=2X

ð@�aÞ �c i��5c i þ

g2scg

32�2f3=2X

aGa��

~Ga�� þ e2c�

32�2f3=2X

aF��~F�� þ . . . ; (2.2)

where F��, Ga��, and Ga

�� are the field strengths respec-tively associated with the Uð1ÞEM, SUð3Þ color, and Ggauge groups; ~F��, ~Ga

��, and~Ga�� are their respective

duals; e, gs, and gG are the respective coupling constantsfor these groups; fX is the fundamental five-dimensionalscale associated with the breaking of the Uð1ÞX symmetry;c�, cg, and ci are coefficients which respectively parame-trize the coupling strength of the five-dimensional axionfield to the photon, gluon, and fermion fields of the SM;and � is anOð1Þ coefficient which depends on the specificsof the axion model in question. Note that in Eq. (2.2), we

have displayed terms involving only the light fields of theSM (i.e., the photon, gluon, and light fermion fields), ascouplings to the heavier SM fields will not play a signifi-cant role in our phenomenological analysis.The five-dimensional axion field can be represented

as a tower of four-dimensional KK excitations via thedecomposition

aðx�; x5Þ ¼ 1ffiffiffiffiffiffiffiffiffiffi2�R

p X1n¼0

rnanðx�Þ cos�nx5R

�; (2.3)

KEITH R. DIENES AND BROOKS THOMAS PHYSICAL REVIEW D 86, 055013 (2012)

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where the factor

rn ��1 for n ¼ 0ffiffiffi2

pfor n > 0

(2.4)

ensures that the kinetic term for each mode is canonically normalized. Substituting this expression into Eq. (2.2) andintegrating over x5, we obtain

Seff ¼Z

d4x

�X1n¼0

�1

2@�an@

�an þ g2G�

32�2fXrnanGa

��~Ga�� þX

i

ci

fXrnð@�anÞ �c i�

��5c i

þ g2scg

32�2fXrnanG

a��

~Ga�� þ e2c�

32�2fXrnanF��

~F��

�� VðaÞ

�; (2.5)

where the axion potential is given by

VðaÞ ¼ X1n¼0

1

2

n2

R2a2n; (2.6)

and where the quantity fX, defined by the relation

f 2X � 2�Rf3X; (2.7)

represents the effective four-dimensional Uð1ÞX-breakingscale. Note that each mode in the KK tower couples to the

SM fields with a strength inversely proportional to fX. Alsonote that at these scales, the axion mass-squared matrix

M 2mn � @2VðaÞ

@am@an

��hai(2.8)

is purely diagonal.At scales T & �G, an additional contribution to the

effective axion potential arises due to instanton effects.In this regime, the potential is modified to

VðaÞ ¼ X1n¼0

1

2

n2

R2a2n þ g2G

32�2�4

G

�1� cos

��

fX

X1n¼0

rnan þ ��G

��; (2.9)

where ��G is the analogue of the QCD theta-parameter ��.This results in a modification of the axion mass-squaredmatrix to

M 2mn ¼ n2M2

c�mn þ g2G�2

32�2

�4G

f2Xrmrn; (2.10)

whereMc � 1=R is the compactification scale. This matrixtakes the form [3]

M 2 ¼ m2X

1ffiffiffi2

p ffiffiffi2

p ffiffiffi2

p. . .ffiffiffi

2p

2þ y2 2 2 . . .ffiffiffi2

p2 2þ 4y2 2 . . .ffiffiffi

2p

2 2 2þ 9y2 . . .

..

. ... ..

. ... . .

.

0BBBBBBB@

1CCCCCCCA;

(2.11)

where

y � Mc

mX

and m2X � g2G�

2

32�2

�4G

f2X: (2.12)

The eigenvalues �2 of this mass-squared matrix are thesolutions to the transcendental equation

��mX

ycot

��

mXy

�¼ �2: (2.13)

The corresponding normalized mass eigenstates a� arerelated to the KK-number eigenstates an via

a� ¼ X1n¼0

U�nan �X1n¼0

�rn ~�

2

~�2 � n2y2

�A�an; (2.14)

where ~� � �=mX. The dimensionless quantity A� is givenby

A� �ffiffiffi2

p~�½1þ ~�2 þ �2=y2��1=2 (2.15)

and obeys the sum rules [3]X�

A2� ¼ 1;

X�

~�2A2� ¼ 1: (2.16)

For T � fX, rewriting Eq. (2.5) in terms of the a� andexpanding the axion potential given in Eq. (2.9) out to

Oða6�=f6XÞ yields the effective action

PHENOMENOLOGICAL CONSTRAINTS ON AXION MODELS . . . PHYSICAL REVIEW D 86, 055013 (2012)

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Seff ¼Z

d4x

�X�

�1

2@�a�@

�a� � 1

2~�2m2

Xa2� þ

e2c� ~�2A�

32�2fXa�F��

~F�� þ g2scg ~�2A�

32�2fXa�G

a��

~G��a

þXi

ci ~�2A�

fXð@�a�Þ �c i�

��5c i

�þ g2G�

4�4G

768�2f4X

X�i;�j;�k;�‘

~�2i~�2j~�2k~�2‘A�i

A�jA�k

A�‘a�i

a�ja�k

a�‘

�: (2.17)

Of course, the interaction term between the a� and the gluon field is only a useful description of the physics at temperaturesabove the quark-hadron phase transition at T ��QCD. At temperatures below this threshold, this interaction term gives riseto an effective Lagrangian containing interactions between the a� and various hadrons, including the proton p, the neutronn, and the charged and neutral pions �� and �0. This Lagrangian takes the form

Lhad ¼ ~�2A�

Ca�

f�fXð@�a�Þ½ð@��þÞ��0 þ ð@��Þ�þ�0 � 2ð@��0Þ�þ�� þ ~�2A�

Can

fXð@�a�Þ �n��5n

þ ~�2A�

Cap

fXð@�a�Þ �p��5pþ i~�2A�

Ca�N

f�fXð@�a�Þ½�þ �p��n� �� n��p�; (2.18)

where the coefficients Ca�, Can, etc., depend on the details of the theory. For example, for a ‘‘hadronic’’ QCD axion [8](i.e., a QCD axion which does not couple directly to the SM quarks), the coefficients Cap and Can, which determine thestrength of the axion-nucleon-nucleon interactions, are

Cap ¼ 0:24

�z

1þ z

�þ 0:15

�z� 2

1þ z

�þ 0:02; Can ¼ 0:24

�z

1þ z

�þ 0:15

�1� 2z

1þ z

�þ 0:02; (2.19)

where z ¼ mu=md � 0:56 is the ratio of the up-quark anddown-quark masses. Likewise, the coefficients Ca�N andCa� for such an axion are

Ca�N ¼ 1� z

2ffiffiffi2

p ð1þ zÞ ; Ca� ¼ 1� z

3ð1þ zÞ ; (2.20)

where f� � 93 MeV is the pion decay constant and m� �135:0 MeV is the neutral pion mass.

Before concluding this review, we note that the effectivecoupling coefficients c�, cg, and ci appearing in Eq. (2.2)

are highly model dependent. They need not beOð1Þ, and inmany theories any of them may vanish outright. Indeed ithas been argued [9] that the existence of axions and axion-like fields which couple to electromagnetism but not toSUð3Þ color is a generic feature of certain extensions of theSM, including string theory. In assessing the constraints onour bulk-axion DDM model, we shall therefore focusprimarily on a ‘‘photonic’’ axion of this sort—i.e., a gen-eral axion with cg ¼ 0 and c� � 0. However, we shall also

discuss how such phenomenological constraints are modi-fied in the case of a so-called ‘‘hadronic’’ axion with non-vanishing values for both cg and c�. We note that

additional subtleties arise in this latter case, due to non-trivial mixings between the an and other pseudoscalarspresent in the theory which also necessarily couple to

Ga��

~Ga��. These include hadrons such as �0 and �, as

well as any other axions in the theory which play a role inaddressing the strong-CP problem [4,5]. In discussingconstraints on hadronic axions, we shall implicitly assumethat the full mass-squared matrix for the theory is such thatthe relationship between the an and the mass eigenstates a�defined in Eq. (2.14) is not significantly disturbed. Indeed,

given the inherently large number of independent scalesand couplings that emerge in scenarios involving multipleaxions and other pseudoscalars, this is not an unreasonableassumption; moreover, it is straightforward to show in ageneral way that these favorable conditions can always bearranged for certain sets of axion and pseudoscalar mix-ings. Such an assumption thereby enables us to perform ourphenomenological analysis in a model-independent way.

III. AXION PRODUCTION IN THE EARLYUNIVERSE

Axions and axionlike fields can be produced via a num-ber of different mechanisms in the early Universe. Forexample, these particles can be produced thermally, viatheir interactions with the SM fields in the radiation bath.In addition, a number of nonthermal mechanisms existthrough which a sizable population of axions also may begenerated. These include production via vacuum misalign-ment, production from the decays of cosmic strings andother topological defects, and production from the out-of-equilibrium decays of other, heavier fields in the theory.This last mechanism is particularly relevant in the contextof the DDM models, since, by assumption, the dark sectorin such models involves large numbers of unstable fieldswith long lifetimes. Indeed, in the axion DDMmodel underconsideration in this paper, a nonthermal population of anya� may be produced via the decays of both heavier KKgravitons and other heavier a�.In Ref. [2], we focused on misalignment production as

the primary mechanism responsible for establishing a cos-mological population of dark axions. In order for theresults for the relic abundances �� of the a� obtained


055013-4

there to be valid, the contributions from all of the alter-native production mechanisms mentioned above must besubdominant for each a�. Therefore, in this section, weexamine each of the relevant axion-production mecha-nisms in turn, beginning with a brief review of the resultsfor misalignment production itself. Since phenomenologi-cal constraints on scenarios involving large, flat extradimensions prefer that the reheating temperature TR asso-ciated with cosmic inflation be quite low [7], we willhereafter operate within the context of a low-tempera-ture-reheating (LTR) cosmology with TR �OðMeVÞ.Within such a cosmological context and within the regionof model-parameter space in which misalignment produc-tion yields a total relic abundance �tot comparable to theobserved dark-matter relic abundance �CDM, we demon-strate that the contributions to each �� from all otherproduction mechanisms are indeed subdominant.

A. Axion production from vacuum misalignment

We begin our discussion of axion production in the earlyUniverse with a brief review of the misalignment mecha-nism and its implications for axion DDM models. (A moredetailed discussion can be found in Ref. [2].) As we shallsee, this mechanism turns out to be the dominant produc-tion mechanism for dark-matter axions in such models.

At temperatures T � �G, the only contributions to theaxion mass-squared matrix are the contributions from theKK masses. Since these contributions to M2 are diagonalin the KK eigenbasis, no mixing occurs, and the KKeigenstates are the mass eigenstates of the theory. Thepotential for each an with n � 0 is therefore nonvanishing,due to the presence of the KK masses, and is minimized atan ¼ 0. However, the potential for the zero mode a0vanishes. In the absence of a potential for a0, there is nopreferred vacuum expectation value (VEV) ha0i whichminimizes Vða0Þ. It therefore follows that immediately

following the phase transition at T � fX, the Universecomprises a set of domains, each with a different homoge-neous background value for the axion field which may be

expressed in terms of a ‘‘misalignment angle’’ �ha0i=fX. This angle is generically expected to be Oð1Þ inany particular domain, but could also be smaller. We

assume here that HI & 2�fX, where HI is the value ofthe Hubble parameter during inflation, and therefore thatthe value of is uniform over our present Hubble volume.In this case, we find that

ha0i ¼ fX; hani ¼ 0 for n � 0: (3.1)

Note that the above discussion is strictly valid only in thelimit in which the Hubble volume is taken to infinity. Inreality, the presence of a finite Hubble volume limits ourability to distinguish fields with wavelengths larger thanthe Hubble radius from true background values. Because ofthis ambiguity, all an for which n=R & HI can also acquireOð1Þ background values after Uð1ÞX breaking. In

Sect. IVL, we will analyze the phenomenological conse-quences of this effect in detail and derive conditions underwhich it can be safely neglected. As we shall demonstrate,it turns out that within our preferred region of parameterspace, these conditions involve only mild restrictions onthe cosmological context into which our model is em-bedded. We will therefore assume from this point forwardthat the hani in our model are given by Eq. (3.1).At temperatures down to T ��G, the hani remain fixed

at these initial values. At lower temperatures, however, thesituation changes as instanton effects generate a potentialfor the axion KK modes. Indeed, in the regime in whichT � �G and the brane mass engendered by this potentialhas attained the constant, low-temperature value mX givenin Eq. (2.12), the time-evolution of each field a� is gov-erned by an equation of the form

€a � þ

t_a� þ �� _a� þ �2a� ¼ 0; (3.2)

where each dot denotes a time derivative, and where

��3=2 in radiation-dominated ðRDÞ eras2 in matter-dominated ðMDÞ eras: (3.3)

When � & 3H=2, the solution to this equation remainsapproximately constant. This implies that the energy den-sity stored in a� scales approximately like vacuum energyduring this epoch. However, at later times, when � *3H=2, we see that a� oscillates coherently around theminimum of its potential, with oscillations damped by a‘‘friction’’ term with coefficient 3H þ ��. During thislatter epoch, the energy density stored in a� scales likemassive matter.At temperatures T ��G, the evolution of a� depends

more sensitively on the explicit time dependence of thebrane massmXðtÞ. In what follows, we adopt a ‘‘rapid-turn-on’’ approximation, in which the instanton potential isassumed to turn on instantaneously at t ¼ tG, where tG isthe time at which the confining transition for the gaugegroup G occurs. In this approximation, mXðtÞ takes theform of a Heaviside step function:

mXðtÞ ¼ mX�ðt� tGÞ: (3.4)

In this approximation, the an remain fixed at the initialvalues given in Eq. (3.1) so long as t < tG. At t ¼ tG, thebrane mass immediately assumes its constant, late-timevalue mX. Since only a0 is populated immediately priorto the phase transition at tG, each of the a� initiallyacquires a background value proportional to its overlapwith a0:

ha�ðtGÞi ¼ fXA�; h _a�ðtGÞi ¼ 0: (3.5)

Subsequently, after the a� have been populated, each be-gins oscillating at a characteristic time scale

t� � max

��

2�; tG

�; (3.6)


055013-5

where � is the value of corresponding to the epochduring which this oscillation begins. At late times t � t�,when these oscillations become rapid compared to the rateof change of ha�i and the virial approximation is thereforevalid, one finds that the energy density �� stored in eachmode is given by

��ðtÞ ¼ 1

22f2X�

2A2�

�t�t

��

e��ðt�tGÞ (3.7)

during the epoch in which the oscillation began.Computing �� during subsequent epochs is simply a matterof applying Eq. (3.7) iteratively with the appropriateboundary conditions at each transition point. Con-sequently, in the LTR cosmology, we have [2]

�LTR� ðtÞ � 1

22f2X�

2A2�e

��ðt�tGÞ

8>>>>>>><>>>>>>>:

t�t

2

t� & t & tRH�t2�

t1=2RH

t3=2

�tRH & t & tMRE�

t2�t1=2MRE

t2t1=2RH

�t * tMRE;

(3.8)

where tRH denotes the reheating time—i.e., the time at which T ¼ TRH, and the Universe transitions from an initial epochof matter domination by the coherent oscillations of the inflaton field to the usual radiation-dominated era.

Given the energy-density expression in (3.8), it is straightforward to obtain the relic abundance�� =�crit for eacha�, where �crit � 3M2

PH2. For the heavier modes in the tower, for which t� ¼ tG, one finds

�LTR� � 3

�fXmX

MP

�2t2G

�1þ �2

m2X

þ �2m2X

M2c

��1e��ðt�tGÞ

8>>>>>><>>>>>>:

14 1=� & t & tRH

49

t

tRH

1=2

tRH & t & tMRE

14

tMRE

tRH

1=2

t * tMRE:

(3.9)

For the modes in the tower for which t� > tG, the corresponding result is

�LTR� � 3

�fXmX

MP

�2��2

�1þ �2

m2X

þ �2m2X

M2c

��1e��ðt�tGÞ

8>>>>><>>>>>:

14 1=� & t & tRH

49

t

tRH

1=2

tRH & t & tMRE

14

tMRE

tRH

1=2

t * tMRE:

(3.10)

The total contribution�tot to the dark-matter relic abun-dance from the axion tower is simply the sum over theseindividual contributions. While the generic behavior of�tot as a function of fX, Mc, and �G is somewhat compli-cated, simple analytical results can be obtained in certainlimiting cases of physical importance. For example, let usconsider the limit in which t� ¼ tG for all modes in thetower andHI is sufficiently large that none of the a� whichwould otherwise contribute significantly to �tot beginoscillating before the end of inflation. In this limit, all ofthe �� take the form given in Eq. (3.9), and one finds thatthe present-day value of�tot, here denoted�

tot, is given by

the simple closed-form expression [2]

�tot � 3

256�2ðgG�Þ2

��2

G

MP

�2t3=2G t1=2MRE

�tGtRH

�1=2

: (3.11)

In the opposite limit, when all of the modes which con-tribute significantly toward�

tot begin oscillating at t� > tG

and have oscillation-onset times which depend on � and aretherefore staggered in time,�� is given by Eq. (3.10) for alla�. In this limit, the expression for �

tot reduces to [2]

�tot � 3

8

�fXMP

�2�tMRE

tRH

�1=2

: (3.12)

The preferred region of parameter space from the per-spective of dark-matter phenomenology is that withinwhich �

tot represents an Oð1Þ fraction of the dark-matterrelic abundance inferred from WMAP data [10]:

�CDMh2 ¼ 0:1131� 0:0034; (3.13)

where h � 0:72 is the Hubble constant. From a dynamicaldark-matter perspective, it is also preferable that the fullaxion tower contribute meaningfully to �

tot. For an Oð1Þvalue of the misalignment angle and a reheatingtemperature within the preferred range TR � 4–30 MeVfor theories with large extra dimensions, one finds [2]


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that these two conditions are realized for fX �1014–1015 GeV and �G � 102–105 GeV, provided thatMc is small enough that y & 1. Within this region ofparameter space, the t� of all a� which contribute mean-ingfully toward �

tot are staggered in time, and thereforethe lighter modes yield a proportionally greater contribu-tion to that total abundance. We will often focus ourattention on this particular region of parameter spacewhen discussing constraints on axion DDM models.

B. Axion production from particle decays

Another mechanism by which a nonthermal populationof relic particles may be generated in the early Universe isthrough the decays of heavier, unstable relics. In scenariosinvolving extra dimensions, these relics include the higherKK modes of any fields which propagate within at leastsome subspace of the extra-dimensional bulk. For example,since the graviton field necessarily propagates throughoutthe entirety of the bulk, a population of unstable KKgravitons is a generic feature of all such scenarios. In theminimal bulk-axion DDMmodel under consideration here,the unstable relics whose decays can serve as a source forany given a� include these KK gravitons as well as other,heavier a�. Moreover, since these fields span a broad rangeof masses from the sub-eV to multi-TeV scale and beyond,one would expect the population of axions produced bytheir collective decays to possess a highly nontrivial phase-space distribution. However, as we shall demonstrate be-

low, the total contribution �ðIEÞ to the decay rate of any a�from intraensemble decays (i.e., decays to final stateswhich include one or more dark-sector fields in additionto any visible-sector fields that might also be present) is farsmaller than that from decays to final states involvingvisible-sector fields alone. That the total branching fractionfor intraensemble decays is negligible suggests that thepopulation of axions produced by such decays will, ingeneral, be quite small. Thus, provided the initial abun-dances of the a� are set by some mechanism such asvacuum misalignment for which the �� of the heavier a�are initially similar to or smaller than those of the lightfields, it is reasonable to assume that the contributions fromintraensemble decays are subleading and may therefore besafely neglected.

One class of processes which contribute to �ðIEÞ arethose which arise due to the axion self-interactions impliedby the final term in Eq. (2.17). The leading such contribu-tion comes from three-body decay processes of the forma� ! a�1

a�2a�3

. An upper bound on the total contribution

�ða� ! 3aÞ to the decay width of a given a� from allkinematically allowed decays of this form was derived inRef. [2]:

�ða� ! 3aÞ � g4G�8

45ð4�Þ7�4

M3c

��G

fX

�8: (3.14)

It was also shown in Ref. [2] that the partial width of the a�to a pair of photons is given by

�ða� ! ��Þ ¼ G�ð~�2A�Þ2 �3

f2X; (3.15)

with G� � c2��2=256�3, where � � e2=4� is the fine-

structure constant. Within the preferred region of parame-

ter space discussed above, in which fX � 1014–1015 GeVand �G * 102–105 GeV, we see that ��ða ! 3aÞ is neg-ligible compared to �ða� ! ��Þ. It then follows that��ða ! 3aÞ represents a vanishingly small contributionto the total decay width �� of any a� in any theory withan Oð1Þ value of c�. We therefore conclude that decays of

the form a� ! a�1a�2

a�3do not play a significant role in

the phenomenology of realistic bulk-axion models of dy-namical dark matter.In addition to these decays, however, an additional set of

decay channels—those involving lighter graviton or radionfields in the final state—are also open to the a�. In order toassess whether such decay channels are capable of yieldinga significant contribution to the relic abundance of any ofthe a�, we begin by identifying the relevant interactionsamong the modes in the KK-graviton and axion towers.Since we are considering the case of a flat extra dimensionand assuming fluctuations of the metric to be small, it isjustified to work in the regime of linearized gravity. Therelevant term in the five-dimensional action is therefore

S ¼ �Z

d4xZ 2�R

0dy

1

M3=25

TMNhMN; (3.16)

where TMN is the stress-energy tensor, and hMN is themetric perturbation defined according to the relation

gMN ¼ �MN þ 2

M3=25

hMN: (3.17)

The piece of the stress-energy tensor which involves thefive-dimensional axion field a includes both a bulk con-tribution and a contribution arising from terms in theinteraction Lagrangian which involve interactions of theaxion with the brane-localized fields of the SM. The bulkcontribution is given by

TbulkMN ¼ @Ma@Na� 1

2�MNð@Pa@PaÞ: (3.18)

Upon KK decomposition, this contribution, when coupledto hMN as in Eq. (3.16), gives rise to three-point interac-tions between a KK graviton or radion field and a pair of

a�. These interactions lead to decays of the form a� !GðmÞ

��a�0 , where GðmÞ�� denotes a KK graviton with KK mode

number m. In the absence of an instanton-induced branemass term mX for the axion field (i.e., in the mX ! 0 limit,in which all mixing between the axion KK modes vanishesand a� ! an), KK-momentum conservation would implythat only a single, marginal decay channel would exist foreach a�. Hence the contribution to �� from such decays


055013-7

can be neglected. However, the instanton contribution tothe axion mass-squared matrix violates KK-momentumconservation, and therefore, despite the fact that theseaxion-axion-graviton interactions are Planck-suppressed,they can still potentially contribute significantly to ��,

due to the large number of modes into which each a� candecay.By contrast, the brane-localized contribution, which is

given by

TbraneMN ¼ �ðyÞ��

M��N

�1

2

Xi

ci

f3=2X

½ð@�aÞ �c i��5c i þ ð@�aÞ �c i��

5c i � 2��ð@�aÞ �c i��5c i�

þ c�e2

32�2f3=2X

að4 ~F��F��

~F� F� Þ þ �g2s

32�2f3=2X

að4 ~Ga��G

�a� � ��

~G� aGa� Þ

�; (3.19)

leads to four-, five-, and six-point interactions between thegraviton field, the a�, and the various SM fields. Theseinteractions take the same form as the interactions whichfollow from the action given in Eq. (2.17) involving theaxion and SM fields alone, save that each vertex involvesthe coupling of an additional KK graviton and is sup-pressed by an additional factor of MP. The rates for suchinteractions will therefore always be much smaller than therates for the corresponding interactions without the addi-tional graviton. Indeed, even the total contribution to thedecay rate of a given a� from such processes, summed overgraviton KK modes, will still be suppressed by a factor ofroughly M5, where M5 denotes the five-dimensionalPlanck scale, relative to the contribution from decays toSM fields alone. It is therefore sufficient, at least for ourpresent purposes, to neglect Tbrane

MN and to focus solely onthe interactions arising from the bulk contribution Tbulk

MN .We begin our analysis of axion-axion-graviton interac-

tions by expanding the five-dimensional axion field, aswell as the various components h��, h�5, and h55 of the

metric perturbation hMN , in terms of KK modes. The mode

expansion of the axion field for the orbifold compactifica-tion considered here was given in Eq. (2.3); the modeexpansions of h��, h�5, and h55 are analogously given by

h�� ¼ 1ffiffiffiffiffiffiffiffiffiffi2�R

p X1m¼0

rmhðmÞ�� cos

�my

R

�

h�5 ¼ 1ffiffiffiffiffiffiffiffiffiffi2�R

p X1m¼1

rmhðmÞ�5 sin

�my

R

�

h55 ¼ 1ffiffiffiffiffiffiffiffiffiffi2�R

p X1m¼0

rmhðmÞ55 cos

�my

R

�:

(3.20)

Note in particular that h�5 must be odd with respect to the

parity transformation x5 ! �x5. Upon substituting theseKK-mode decompositions into the linearized-gravity ac-tion given in Eq. (3.16) and integrating over y, we find thatthe terms in the effective, four-dimensional interactionLagrangian which govern the interactions between thegraviton and axion KK modes consist of the followingthree contributions:

Z 2�R

0

h��T��bulk

M3=25

dy ¼ X1m;n;p¼0

rmrnrp4MP

hðmÞ��

�ð2@�aðnÞ@�aðpÞ � ��@�aðnÞ@�aðpÞÞ�þ

mnp þ ��

�np

R2

�aðnÞaðpÞ��

mnp

�Z 2�R

0

h55T55bulk

M3=25

dy ¼ X1m;n;p¼0

rmrnrp4MP

hðmÞ55

�@�aðnÞ@�aðpÞ�þ

mnp þ�np

R2


mnp

�

Z 2�R

0

h�5T�5bulk

M3=25

dy ¼ X1m¼1

X1n;p¼0

rmrnrp2MP

�p

R

�hðmÞ�5 ð@�aðnÞÞaðpÞ��

nmp;

(3.21)

where

��mnp � ½�m;n�p þ �m;p�n� � ½�m;nþp þ �m;�n�p�:

(3.22)

For the purposes of computing Feynman diagrams, it is

convenient to work in the unitary gauge, in which the hðmÞ�5

and hðmÞ55 fields with m> 0 are set to zero by the five-

dimensional gauge transformations gMN ! gMN þ@M�N þ @N�M, where �M is the gauge parameter. In this

gauge, the contributions in the second and third lines ofEq. (3.21) vanish (save for the interactions between the

axion KK modes and the radion field hð0Þ55 , which will be

discussed in due time), and the physical, gauge-invariantgraviton fields

GðmÞ�� hðmÞ

�� þ�R

m

�½@�hðmÞ

�5 þ @�hðmÞ�5 � �

�R2

m2

�@�@�h

ðmÞ55

(3.23)


055013-8

reduce to hðmÞ�� for all m> 0. The relevant part of the effective Lagrangian consequently reduces to

L ðm>0Þint ¼ � X1

m¼1

X1n;p¼0

rnrp

2ffiffiffi2

pMP

hðmÞ��

�ð2@�aðnÞ@�aðpÞ � ��@�aðnÞ@�aðpÞÞ�þ

mnp þ ��

�np

R2


mnp

�: (3.24)

The expression in Eq. (3.24) can be rewritten in terms of the mass eigenstates a� via the mixing matrix U�n in Eq. (2.14).The result is

Lðm>0Þint ¼ � X1

m¼1

X1n;p¼0

X�;�0

rnrp

2ffiffiffi2

pMP

hðmÞ��U

yn�U

yp�0

�ð2@�a�@�a�0 � ��@�a�@�a�0 Þ�þ

mnp þ ��

�np

R2

�a�a�0��

mnp

�

¼ � X1m¼1

X1n¼0

X�;�0

rn

2ffiffiffi2

pMP

hðmÞ��U

yn�

�ð2@�a�@�a�0 � ��@�a�@�a�0 Þ

ðrn�mUyn�m;�0 þ rnþmU

ynþm;�0 þ rm�nU

ym�n;�0 þ r�n�mU

y�n�m;�0 Þ þ �� n

R2a�a�0

�ðn�mÞðrn�mU

yn�m;�0 þ rm�nU

ym�n;�0 Þ þ ðnþmÞðrnþmU

ynþm;�0 þ r�n�mU

y�n�m;�0 Þ

��; (3.25)

where in going from the first equality to the second we have exploited the Kronecker deltas in ��mnp to evaluate the sum

over p. It should be noted that in the notation employed in the above expression,Uyn� ¼ 0 by definition for n < 0. The sum

over n in Eq. (3.25) can also be performed analytically, and the resulting, final expression for the Lagrangian in terms of thea� is found to be

L ðm>0Þint ¼ � X1

m¼1

X�;�0

1

2ffiffiffi2

pMP

hðmÞ�� ½ð2@�a�@�a�0 � ��@�a�@�a�0 ÞCð1Þ

m��0 þ ��M2ca�a�0Cð2Þ

m��0 �; (3.26)

where the coefficients Cð1Þm��0 and Cð2Þ

m��0 are given by

Cð1Þm��0 ¼ �8m2y2 ~�2 ~�02A�A�0

m4y4 � 2m2y2ð~�2 þ ~�02Þ þ ð~�2 � ~�02Þ2 ; Cð2Þm��0 ¼ 4~�2 ~�02½m2y2ð~�2 þ ~�02Þ � ð~�2 � ~�02Þ2�A�A�0

y2½m4y4 � 2m2y2ð~�2 þ ~�02Þ þ ð~�2 � ~�02Þ2� : (3.27)

From the interaction Lagrangian in Eq. (3.26), it is straightforward to obtain the Feynman rule for the graviton-axion-axion interaction vertex in the unitary gauge:

Using this vertex rule along with the graviton-polarization sum rule given in Refs. [11,12], we find

jMða� ! hðmÞ��a�0 Þj2 ¼ ðCð1Þ

m��0 Þ212M2

PðmMcÞ4½�4 þ ðmMcÞ4 þ �04 � 2ðmMcÞ2�2 � 2ðmMcÞ2�02 � 2�2�02�2

¼ 16

3

�m4

X

M2P

�ð~�2A�Þ2ð~�02A�0 Þ2 (3.28)

for �0 < �. Consequently, the partial width of a� from such a decay is


055013-9

�ða� ! hðmÞ��a�0 Þ ¼ m4

X

3��3M2P

ð~�2A�Þ2ð~�02A�0 Þ2½�4 þ ðmMcÞ4 þ �04 � 2ðmMcÞ2ð�2 þ �02Þ � 2�2�02�1=2: (3.29)

Once again, in order to obtain the full contribution �ða� ! h��aÞ to �� from decays of the form a� ! hðmÞ��a�0 , it is

necessary to sum over all combinations of final-state graviton and axion modes which are kinematically accessible. Asbefore, we will approximate the mode sums over both m and �0 as integrals. This yields the result

�ða� ! h��aÞ & 4m4Xð~�2A�Þ2

3��3McM2P

Z �

�0

d�0 Z ð��0Þ=Mc

0dmð~�02A�0 Þ2

½�4 þ ðmMcÞ4 þ �04 � 2ðmMcÞ2ð�2 þ �02Þ � 2�2�02Þ�1=2

¼ 8m4Xð~�2A�Þ2

9��3M2cM

2P

Z �

�0

d�0ð~�02A�0 Þ2ð�þ �0Þ�ð�2 þ �02ÞE

�ð�� 0Þ2ð�þ �0Þ2

�� 2��0K

�ð�� 0Þ2ð�þ �0Þ2

��; (3.30)

where KðxÞ and EðxÞ denote the complete elliptic integrals of the first and second kind, respectively:

KðxÞ ¼Z �=2

0

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2sin2

p ; EðxÞ ¼Z �=2

0d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2sin2

p: (3.31)

In order to compare �ða� ! h��aÞ to the rate for a�decays to SM fields, we numerically integrate Eq. (3.30)over �0 and compare the resulting expression to the decayrate �ða� ! ��Þ to photon pairs. In Fig. 1, we plot theratio �ða� ! h��aÞ=�ða� ! ��Þ as a function of � for avariety of different choices of fX. In each case, we have set�G ¼ 1 TeV, Mc ¼ 10�11 GeV, and � ¼ gG ¼ 1. It isevident from this plot that only for values of fX abovethe preferred range fX � 1014–1015 GeV does the decay

rate for a� ! h��a�0 become similar in magnitude to therate for axion decays into brane fields. Indeed, for valuesfX within this preferred range, �ða� ! h��aÞ=�ða� !��Þ never exceeds 0.06, even for the lightest modes inthe tower. Furthermore, for values of fX of this magnitude,the lifetimes for all a� light enough to have �ða� !h��aÞ=�ða� ! ��Þ near this maximal value are parametri-cally larger than the present age of the Universe, even oncethe additional contribution to �� from a� ! h��a�0 decaysis taken into account. Consequently, the decays of suchfields are not cosmologically relevant, and the branchingfraction for all other, heavier a� into final states involvingKK gravitons is utterly negligible. We therefore concludethat intraensemble decays are not phenomenologicallyrelevant for bulk-axion models of dynamical dark matter.Up to this point, we have focused chiefly on the effect of

the tensor KK modes of the higher-dimensional gravitonfield on axion production in the early Universe. However,we have yet to address the effect of graviscalars such as theradion on axion production. Since our minimal DDMmodel involves only a single extra dimension, only a single

physical graviscalar mode (proportional to hð0Þ55 ) appears in

the theory. Furthermore, while the masses of the hðmÞ�� are

dictated by the compactification geometry alone, the massof this radion field depends on the details of the mechanismthrough which the radius of the extra dimension is stabi-lized, and is consequently highly model dependent. In thispaper, we assume that the physical radion field is suffi-ciently heavy so as not to play a significant role in thedecay phenomenology of the light a� fields which contrib-ute significantly to �

tot. Nevertheless, we note that inscenarios which involve multiple extra dimensions of com-parable size, or scenarios in which a specific model forradius stabilization is invoked, graviscalars may play a

f X 1011 GeV

f X 1012 GeV

f X 1013 GeV

f X 1014 GeV

f X 1015 GeV

G 1 TeV

10 12 10 9 10 6 0.001 110 12

10 9

10 6

0.001

1

GeV

ah

a'

a

FIG. 1 (color online). The ratio �ða� ! h��aÞ=�ða� ! ��Þ,shown as a function of � for several different values of fX. Herewe have set �G ¼ 1 TeV and � ¼ gG ¼ 1, and we have takenthe compactification scale to be Mc ¼ 10�11 GeV. It is clearfrom this plot that this ratio is safely below unity for fX withinour preferred region 1014–1015 GeV.


055013-10

more significant role in the phenomenology of the darksector.

C. Axion production from cosmic strings

A population of cold axions can also be generated by thedecays of topological defects. In our axion DDM model,this includes decays of the cosmic strings associated withthe breaking of the global Uð1ÞX symmetry. Such decaysare relevant in situations in which this symmetry remains

unbroken until after inflation, i.e., HI * 2�fX. By con-

trast, in situations in which HI & 2�fX and the Uð1ÞX isspontaneously broken prior to the inflationary epoch, cos-mic strings and other topological defects are washed out bythe rapid expansion of the Universe during cosmic infla-tion. Consequently, in this latter case, axion productionfrom the decays of cosmic strings can safely be ignored.

In this paper, we are primarily interested in high values

of fX � 1014–1015 GeV, as these values characterize ourpreferred region of parameter space. Likewise, we willprimarily be interested in relatively low values of HI,which may be realized naturally in the LTR cosmology.

For this reason we shall assume that HI & 2�fX in whatfollows. We see, then, that no significant population ofaxions is produced by cosmic-string decay.

D. Axion production from the thermal bath

Another mechanism through which a relic population ofaxions may be produced in the early Universe is a thermalone: via their interactions with the SM fields in the radia-tion bath. Unlike the axion population generated by vac-uum misalignment, which is characterized by a highly

nonthermal velocity distribution (essentially that of aBose-Einstein condensate) and is therefore by naturecold, this population is characterized by a thermal velocitydistribution. Indeed, the properties of a thermal populationof axions can differ substantially from that of a populationof axions generated via misalignment production.A number of processes contribute to thermal axion

production in the early Universe, and the processes whichare the most relevant for the production of standard axionsdominate for each a� in this scenario as well. Amonghadronic processes, which play an important role in axionproduction when cg is nonvanishing, q� ! qa� and qg !qa� dominate for T * �QCD, while pion-axion conversion

off nuclei (including all processes of the formN� ! N0a�,where N, N0 ¼ fn; pg and � denotes either a charged orneutral pion) and the purely pionic process �� ! �a�dominate at lower temperatures. The rate for the high-temperature process is [13]

�ðq� ! qa�Þ ¼ g2sT3 ~�4A2

�

64�5f2Xln

��T

mg

�2 þ 0:406

�; (3.32)

where mg is the plasma mass for the gluon, given in terms

of the effective number of quark flavors Nf at temperature

T by

mgðTÞ ¼ gsT

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ Nf=2

q: (3.33)

Likewise, the rates for pion-conversion off nuclei andpionic production are well estimated by the expressions[14–16]

�ðN� ! N0a�Þ ¼ T7=2m3=2N

~�4A2�e

�mN=T

6�ð3Þð2�Þ5=2f2Xf2�½1:64ð5C2

an þ 5C2ap þ 2CanCapÞ þ 6C2

a�N�Z 1

0dx1

x1y31

ey1 � 1

�ð�� ! �a�Þ ¼ 3�ð3ÞT5C2a�

~�4A2�

1024�7f2Xf2�

Z 1

0

Z 1

0

dx1dx2x21x

22

y1y2ðey1 � 1Þðey2 � 1ÞZ 1

�1d�

ðs�m2�Þ3ð5s� 2m2

�Þs2T4

;

(3.34)

where, once again, �ðxÞ denotes the Riemann zeta function,and the effective coupling coefficients Cap, Can, Ca�, andCa�N are given in Eqs. (2.19) and (2.20). Since theseprocesses are mediated by strong interactions, they tendto dominate the production rate for a hadronic axionat temperatures T * 100 MeV, at which the numberdensities of pions and other hadronic species areunsuppressed.

In addition to these hadronic processes, there are severalprocesses involving the interactions between the a� and thee� and photon fields which contribute to the axion-production rate, and indeed dominate that rate at tempera-tures T � �QCD. The first of these is the inverse-decay

process �� ! a�, the rate for which is given by

�ð�� ! a�Þ ¼ 2�5G�ð~�2A�Þ2�ð3Þf2XT2

K1

��

T

�; (3.35)

where K1ðxÞ denotes the Bessel function of the first kind,and G� ¼ �2c2�=256�

2. Another is the Primakoff process

e�� ! e�a. For T, me � �, the rate for this process iswell approximated by [17]

�Primðe��! e�a�Þ¼�3c2�ne

192�ð3Þf2X~�4A2

�

�ln

�T2

m2�

�þ0:8194

�;

(3.36)

where the plasma mass m� of the photon is given by

m� ¼ eT=3. In the approximation of vanishing chemical


055013-11

potential, the number density of electrons (plus positrons)ne takes the well-known form

ne ¼8><>:

3�ð3Þ�2 T3 T * me

4Tme

2�

3=2

e�me=T T & me:(3.37)

Finally, if ce � 0 in Eq. (2.5) and the axion couples directlyto the electron field, there can be an additional contributionto the e�� ! e�a rate from a process akin to Comptonscattering, but with an axion replacing the photon in thefinal state. The rate for this process can be estimated as [18]

�Compðe�� ! e�a�Þ � 4�c2ene

f2Xð~�2A�Þ2

(m2

e

T2 T * me

1 T & me:

(3.38)

In Fig. 2, we provide a pictorial comparison of the ratesfor the axion-production processes enumerated above asfunctions of temperature. The left panel shows the rates forthe production of a single axion species a� with � ¼1 MeV in a scenario with fX ¼ 1015 GeV, Mc ¼10�11 GeV, and �G ¼ 1 TeV. As before, we have taken� ¼ 1 and set cg ¼ c� ¼ ce ¼ 1. The red curve corre-

sponds to the rate �ð�� ! �a�Þ for the pionic process;the orange curve to the rate �Primðe�� ! e�a�Þ for thePrimakoff process; the green curve to the rate �ð�� ! aÞfor the inverse-decay process; the blue curve to the rate

�ðN� ! N0a�Þ for the pion-conversion process off nuclei;and the purple curve to the rate �ðq� ! qa�Þ for thequark-gluon process. As the hadron description of thetheory is valid only for T & �QCD, and likewise,

the quark/gluon description is only valid for T * �QCD,

the rates �ð�� ! �a�Þ, �ðN� ! N0a�Þ, and �ðq� !qa�Þ are only defined on one side or the other of this scale.The yellow curve corresponds to the rate �Compð�� !�a�Þ for the Compton-like process for ce ¼ 1. For pur-poses of comparison, we also show the Hubble parameteras a function of T for two different cosmologies: thestandard cosmology (black dashed curve), and an LTRcosmology with a reheating temperature TRH ¼ 5 MeV(black dot-dashed curve). The value of � we have chosenhere is well within the asymptotic regime for this choice of

Mc and fX; hence the rates displayed here represent themaximal values possible for any a� in the scenario. In theright panel of Fig. 2, we show, for the same choice of Mc

and fX, the total contribution to the axion-production rateobtained by summing the rates for all a� for which� � T—i.e., those which will be kinematically accessibleat a given temperature.The most salient lesson to draw from Fig. 2 is that even

after the contributions from all kinetically accessible a�states are included in the thermal axion-production rate,none of the relevant processes by which a thermal popula-tion of axions might be produced comes close to satisfying

aPrim e eaComp e ea

aqg qaN N'a

HStd

HLTR

10 7 10 5 0.001 0.1 1010 60

10 51

10 42

10 33

10 24

10 15

T GeV

int

GeV

c 1, cg 0c 1, cg 1

HStd

HLTR

10 7 10 5 0.001 0.1 1010 60

10 51

10 42

10 33

10 24

10 15

T GeV

int

GeV

FIG. 2 (color online). A comparison of the rates associated with different axion-production processes in the early Universe. Here wehave taken Mc ¼ 10�11 GeV, fX ¼ 1015 GeV, �G ¼ 1 TeV, TRH ¼ 5 MeV, and � ¼ 1. The left panel shows the production rate foreach process for an individual axion species a� with � ¼ 1 MeV (i.e., a value well within the asymptotic, large-� regime, where therates are the least suppressed). The right panel shows the integrated production rate for each process, including contributions from allmodes with � < T. The most relevant processes for thermal axion production in this scenario are �� ! �a� production (red curve),e�� ! e�a� via the Compton process (yellow curve), e�� ! e�a� production via the Primakoff process (orange curve), inversedecays of the form �� ! a� (green curve), production via the quark-gluon process qg ! qa (blue curve), and pion-production offnuclei (purple curve). It should be noted that the Compton process requires a nonzero electron-electron-axion coupling ce, and that thecurve shown here corresponds to the case in which ce ¼ 1. The value of the Hubble parameter as a function of T in both the standard(black dashed curve) and LTR (black dash-dotted curve) cosmologies are also shown.


055013-12

the ��H criterion. This implies that the a�, even whentaken together, never attain thermal equilibrium with theplasma after inflation ends. Furthermore, these results alsojustify the claims made above, that the electron Primakoffprocess and inverse decays of the form �� ! a are themost relevant axion-production processes for T & �QCD,

while hadronic processes dominate the axion-productionrate for T * �QCD.

Let us now estimate the contribution to �tot from ther-mal axion production in the context of an LTR cosmologywith a reheating temperature of TRH ¼ 5 MeV. For con-creteness, we focus on the case of a photonic axion withc� ¼ 1 and cg ¼ ci ¼ 0 for all i; however, the results for

other coupling assignments should not differ drasticallyfrom those obtained here. We begin by noting that anycontribution to �� generated at temperatures T * TRH,i.e., during the reheating phase, will be substantially diluteddue to entropy production from inflaton decays. It is there-fore legitimate to restrict our attention to axion productionwithin the subsequent RD era. For a photonic axion, theprocesses which contribute to thermal axion production areinverse decays and e�� ! e�a, the latter of which, sincewe are assuming ce ¼ 0, is dominated by the Primakoffprocess. The Boltzmann equation for the number densityn� of each a� is therefore effectively given by

_n � þ ð3H þ ��Þn� ¼ CID� ðTÞ þ CPrim

� ðTÞ (3.39)

for T & TRH, where CPrim� ðTÞ and CID

� ðTÞ are the contactterms associated with the electron-Primakoff and inverse-decay rates given in Eqs. (3.36) and (3.35), respectively. ForT � �, me, these contact terms are well approximated bythe expressions

CPrim� ðTÞ � 2�

3�2G�ð~�2A�Þ2 T

6

f2X

�ln

�9

4��

�þ 0:8194

�

CID� ðTÞ � 2G�ð~�2A�Þ2 �5T

�2f2XK1

��

T

�; (3.40)

where K1ðxÞ denotes the Bessel function of the first kind.To obtain a rough estimate of the relic abundance insituations in which either me or � is comparable to orgreater than T, we modify the expression forCPrim

� ðTÞ givenin Eq. (3.40) by including an additional exponential factor

e�ð�þmeÞ=T to model the effect of Boltzmann suppression.From Eq. (3.39) we estimate the relic abundance of

axions produced by interactions with the SM particles inthe thermal bath. To do so, we neglect the decay term andrewrite the resulting equation in terms of the quantity Y� �n�=s, where s is the entropy density, in order to remove theHubble term:

s _Y� � CID� ðTÞ þ CPrim

� ðTÞe�ð�þmeÞ=T: (3.41)

By numerically integrating this equation, we obtain an

estimate of the thermal contribution �ðthermÞ� to the abun-

dance �� of each a� at present time:

�ðthermÞ� � �T3

nowtMRE

�crit

Z TRH

Tnow

3

ðTÞ�TMRE

T

�3=ðTÞ gsðTnowÞ

gsðTÞ ½CID� ðTÞ þ CPrim

� ðTÞe�ð�þmeÞ=T�dT; (3.42)

where gsðTÞ is the number of interacting degrees of free-dom present in the thermal bath at temperature T, andwhere ðTÞ is defined in Eq. (3.3). The results of thisintegration are displayed in Fig. 3. In this figure, wecompare the contributions to the relic abundance �� of agiven a� from misalignment production and thermal pro-duction for a variety of different choices of the modelparameters.

It is clear from Fig. 3 that for these parameter assign-

ments, �ðthermÞ� only becomes comparable with the relic-

density contribution �ðmisÞ� from vacuum misalignment for

reasonably heavy a�. Neither �ðmisÞ� nor �ðthermÞ

� for such

a� is non-negligible compared with the�ðmisÞ� contribution

from the lighter modes. Indeed, summing over � to obtain

the total thermal contribution �ðthermÞtot to the axion relic

abundance at present time yields 3:8 10�6 & �ðthermÞtot &

3:8 10�4. We may therefore safely conclude that

�ðthermÞtot � �ðmisÞ

tot within the preferred region of parameterspace for bulk-axion models of dynamical dark matter, andthat the population of a� generated by the misalignment

mechanism dominates the relic density of the DDMensemble.To summarize the results of this section, we have exam-

ined the primary mechanisms through which a cosmologi-cal population of DDM axions may be generated, includingmisalignment production, thermal production, and produc-tion by decaying relics. We have shown that within thepreferred region of parameter space specified in Ref. [2],the contribution to the total present-day dark-matter relic

abundance from misalignment production �ðmisÞtot indeed

dominates over the contributions from all other productionmechanisms. This justifies the emphasis placed on mis-alignment production in Ref. [2]. Still, we note thatalthough populations of axions produced via those otherchannels collectively represent a negligible fraction of�tot, those populations can nevertheless play an importantrole in constraining bulk-axion DDM models. For ex-ample, the thermal population of axions discussed abovecan still leave a significant imprint on the diffuse X-ray

spectrum despite the small size of �ðthermÞtot , because

�ðthermÞ� � �ðmisÞ

� when � is large. We shall return to this


055013-13

point in Sect. IVG, where we will show that this imprint isnevertheless consistent with current observational limits.

IV. PHENOMENOLOGICAL CONSTRAINTS ONDARK AXION ENSEMBLES

In the previous section we characterized the variousmechanisms which contribute to the generation of a cos-mological population of relic axions in axion DDMmodelsand compared the sizes of their contributions to�

tot. Giventhat this population constitutes the dark-matter ensemble inour axion DDM model, we now turn to examine therelevant phenomenological, astrophysical, and cosmologi-cal constraints on that population of axions. As we shallsee, some of these constraints pertain generically to anytheory of dark matter, or to any theory containing late-decaying relics. Others are particular to models involvinglight, weakly coupled fields. Still others pertain to theorieswith large extra dimensions in general, regardless of thepresence or absence of a bulk-axion field.

As we have seen in Refs. [1,2], the properties of thedark-matter ensemble and its constituent fields in ourbulk-axion DDM model are determined primarily bythree parameters: the compactification scale Mc, the

Uð1ÞX-breaking scale fX, and the confinement scale �G

for the gauge groupG. Because these parameters play sucha central role in characterizing the dark sector in ourmodel, we shall seek to phrase our phenomenological

constraints in terms of restrictions on Mc, fX, and �G

whenever possible. Of course, in addition to these primaryparameters, a number of other ancillary quantities alsohave an impact on the phenomenology of our model, and

thus are also constrained by data. These include the scalesHI and TRH associated with cosmic inflation, the couplingcoefficients cg, c�, and ci, and so forth. Generally speak-

ing, these additional parameters play a subordinate role indetermining the mass spectrum and relic abundances of thea�, and the values they take are typically far more model

dependent than Mc, fX, and �G. Thus, while certain ex-perimental and observational limits serve to constrain thevalues these additional parameters may take, it ultimatelyturns out to be possible to phrase the majority of con-

straints on our model as bounds on Mc, fX, and �G.Indeed, as we shall see in Sect. V, most of the criticalbounds can be expressed conveniently in this manner. Wewill also be interested in how these bounds constraincertain derived quantities of physical importance, such asthe quantity y defined in Eq. (2.12), which quantifies theamount of mixing that occurs across our DDM ensemble.

A. Constraints from background geometry

The first set of constraints we consider are those whichapply generically to theories with extra dimensions, inde-pendently of the presence or properties of the bulk-axionfield whose KK excitations constitute the DDM ensemblein our model. These constraints arise primarily from ex-perimental limits on the physical effects to which the towerof KK gravitons necessarily present in such theories givesrise. We will primarily focus here on scenarios involving nflat extra dimensions in which the fields of the SM arelocalized on a 3-brane, while gravity, as always, neces-sarily propagates throughout the entirety of the D ¼ ð4þnÞ-dimensional bulk.

f X 1012 GeV

f X 1013 GeV

f X 1014 GeV

f X 1015 GeV

Mc 10 11 GeVG 1 TeV

10 12 10 9 10 6 0.001 110 35

10 27

10 19

10 11

0.001

105

GeV

f X 1012 GeV

f X 1013 GeV

f X 1014 GeV

f X 1015 GeV

Mc 10 11 GeVG 100 TeV

10 12 10 9 10 6 0.001 110 35

10 27

10 19

10 11

0.001

105

GeV

FIG. 3 (color online). Contributions to the individual mode abundances �� for a photonic axion from thermal production (dashedcurves) and misalignment production (solid curves), plotted as functions of � for fX ¼ 1012 GeV (red curves), fX ¼ 1013 GeV(orange curves), fX ¼ 1014 GeV (green curves), and fX ¼ 1015 GeV (blue curves). The left panel displays the results for �G ¼1 TeV, while the right panel displays the results for �G ¼ 100 TeV. The other model parameters have been set to Mc ¼ 10�11 GeV,TRH ¼ 5 MeV, � ¼ gG ¼ c� ¼ 1.


055013-14

Perhaps the most significant and direct bound on Mc intheories with extra dimensions arises due to modificationsof Newton’s law at short distances as a consequence ofKK-graviton exchange. The lack of evidence for any sucheffect at modified-gravity experiments to date implies con-straints on the sizes and shapes of those extra dimensions.For the case of a single large, flat extra dimension, thecurrent limit on the compactification scale from such ex-periments is [19]

Mc * 3:9 10�12 GeV: (4.1)

This lower limit on the compactification scale is robust inthe sense that even if there exist additional compact di-mensions with radii ri � 1=Mc, this bound is essentiallyunaffected. For this reason, Eq. (4.1) turns out to representthe most significant constraint on the parameter space ofbulk-axion DDMmodels from considerations which derivesolely from the presence of extra dimensions.

There also exist additional constraints on the compacti-fication geometry which arise due to the relationship be-tween this scale, the effective four-dimensional Planckscale MP, and the fundamental scale of quantum gravityMD. These constraints are generally more sensitive to thedetails of the compactification scenario. In general, thefundamental scale MD is related to MP by [6]

M2P ¼ VnM

2þnD ; (4.2)

where Vn is the volume of the n-dimensional manifold onwhich the extra dimensions are compactified. For the sim-ple case in which this manifold is a flat, rectangularn-torus, the volume Vn is simply the product of (2�ri)for each cycle of the torus. Assuming all radii are equal to acommon radius r, we then have

r�1 � 2�MminD

�Mmin

D

MP

�2=n

: (4.3)

Bounds on the scale MD appearing in the literature arefrequently predicated on these assumptions. However, weemphasize that in situations in which the ri are not allequal, or in which the compactification geometry differsfrom that of a flat, rectangular n-torus, those bounds can beconsiderably modified.

Under the assumption that the compactification geome-try resembles that on which Eq. (4.3) is predicated, onemay derive constraints on MD, r, or combinations of thetwo. For example, one class of constraints which arise intheories with extra dimensions are those implied by thenonobservation of effects related to thermal KK-gravitonproduction in astrophysical sources such as stars [20] andsupernovae [21,22]. A brief synopsis of the most relevantbounds in this class is given in Ref. [20], all of whichdepend crucially on the fundamental quantum-gravityscaleMD. The most stringent of these constraints currentlyderives from limits on photoproduction and stellar heatingby gravitationally trapped KK gravitons in the halos of

neutron stars. Indeed, for a theory involving n extra di-mensions with equal radii, one finds that for n ¼ 2, thebound is r�1 � 5:8 10�7 GeV, while for n ¼ 3, onefinds r�1 � 3:8 10�10 GeV [20].Collider data also place limits on r and MD in theories

with extra dimensions. Searches for evidence of KK-graviton production in the monojet (i.e., jþ 6ET) channelhave been performed by the ATLAS [23,24] and CMS [25]collaborations. The most recent ATLAS analysis [24],conducted with 1 fb�1 of integrated luminosity, constrainsMD * f3:16; 2:50; 2:15g TeV at 95% C.L. for n ¼ f2; 3; 4gflat extra dimensions with equal radii. The most recentCMS analysis [25], conducted at a comparable integratedluminosity, yields the slightly more stringent constraintMD * f4:03; 3:21; 2:80g TeV at 95% C.L. for the corre-sponding values of n. Limits from searches for KK-graviton effects in the diphoton [26] and dimuon [27]channels at 36 pb�1 and 39 pb�1 of integrated luminosity,respectively, have also been derived by the CMSCollaboration, but these are currently less stringent thanthe constraints from the jþ 6ET channel.It is important to realize that the aforementioned bounds

on MD as a function of the compactification geometry donot necessarily translate directly into analogous bounds on

fX for a given fX. Unlike the graviton field, the bulk-axionfield in our DDM model need not necessarily propagatethroughout the entirety of the extra-dimensional volume,but may in principle also be confined to a (4þ na)-dimensional subspace of that volume, where na < n.

When this is the case, fX is related to fX by the general-ization of Eq. (2.7):

f 2X ¼ Vnaf

2þnaX : (4.4)

Note that this relationship differs from that which existsbetween MP and MD because na < n. In this paper, as inRef. [2], we focus on the case in which the axion fieldpropagates in a single extra dimension of radius R, irre-spective of the size, shape, or number of extra dimensionswhich compose the totality of the bulk. Accordingly, wedefine Mc ¼ 1=R to be the compactification scale associ-ated with this particular extra dimension, and we shall usethis notation throughout. In this paper, we are not aiming toset MD at or even near the TeV scale, since we are notattempting to solve the hierarchy problem, but rather toaddress the dark-matter problem. We will therefore assumethat the structure of any additional bulk dimensions is suchthat phenomenological constraints on MD and the associ-ated compactification geometry are satisfied. Note, how-ever, that the Newton’s-law bound in Eq. (4.1) does applytoMc, as it applies to the compactification scale associatedwith any individual extra dimension.Another class of constraints on scenarios involving

large extra dimensions applies to ancillary variables whichcharacterize the cosmological context in which our modelis situated. For example, the prediction of the observed


055013-15

abundances of the light elements via big-bang nucleosyn-thesis is one of the greatest successes of the standardcosmology. Consistency with these predictions requiresthat effects stemming from the presence of these extradimensions not disrupt BBN. Successful nucleosynthesisrequires that the expansion rate of the Universe during theBBN epoch, as quantified by the Hubble parameter HðTÞ,must not deviate from its usual, four-dimensional value bymore than around 10% [7]. In other words, there existssome temperature T � TBBN � 1 MeV (usually dubbedthe ‘‘normalcy temperature’’ in the literature) below whichthe radii of all extra dimensions are effectively fixed andthe bulk is effectively empty of energy density. Avariety ofdifferent considerations constrain T, most of which arerelated to the potentially observable effects of KK-gravitondynamics in the early Universe:

(i) Interactions between the SM fields on the brane andthe bulk graviton field result in a transfer of energyfrom the brane to the bulk, and a consequent coolingof the radiation bath on the brane. Substantial energyloss via this ‘‘evaporative cooling’’ mechanismwould result in a modification of the expansion rateof the Universe. At temperatures T & TBBN, such amodification would distort the light-element abun-dances away from those predicted by standard BBN.Thus, the strength of the interactions between SMparticles and excitations of the graviton field isconstrained.

(ii) If the collective energy density associated with thegraviton KK modes is substantial, that energy den-sity could cause the Universe to become matter-dominated too early. In extreme cases, it couldeven overclose the Universe.

(iii) Late decays of KK gravitons could result in dis-tortions of the abundances of light elements awayfrom the values predicted by BBN [7], which ac-cord well with the observed values for these abun-dances. Such decays could also result in significantentropy production.

(iv) The relationship between the Hubble parameter Hand the total energy density � of the Universe ismodified at early times in higher-dimensional sce-narios, even when that energy density is over-whelmingly dominated by brane-localized states[28–30]. Such a modification could have a substan-tial effect on BBN as well.

The constraints on T implied by these considerationshave been reckoned by a number of authors [7,31], andwhile the precise values of the bounds so derived againdepend on the number, size, and shape of the extra dimen-sions, the value ofMD, etc., the most stringent (which tendto come from limits on the late decays of the excited KKmodes) generally tend to restrict T to within the roughrange 4 MeV & T & 20 MeV [7].

One possibility for achieving such conditions is to positthat T be identified with the reheating temperature TRH

associated with a period of cosmic inflation initiated by aninflaton field which is localized on the same 3-brane as theSM fields. During such an inflationary epoch, any contri-butions to the energy density of the Universe from bulkstates which existed prior to the inflationary epoch (savefor those which, like the contributions to �� from vacuummisalignment, scale like vacuum energy) are inflated away.Furthermore, if the inflaton field decays primarily to otherbrane-localized states, no substantial population of bulkstates is regenerated during the subsequent reheatingphase. Thus, by adopting a LTR cosmology with a reheat-ing temperature 4 MeV & TRH & 20 MeV, we therebyensure that the relevant constraints related to KK-gravitonproduction in the early Universe are satisfied. We also notethat a reheating temperature of TRH * 4 MeV is sufficientto ensure that the thermal populations of the SM fields(and, in particular, the three neutrino species) required instandard BBN are generated by the thermal bath afterreheating [32,33].For completeness, it is worth noting that enforcing com-

patibility with an LTR cosmology often constrains otheraspects of the cosmological model unrelated to dark-matterphenomenology. The set of viable mechanisms for baryo-genesis is particularly impacted by such restrictions.Indeed, since a baryon asymmetry generated prior to theend of inflation will effectively be erased during the infla-tionary epoch, many canonical baryogenesis scenarios (in-cluding high-scale leptogenesis) are incompatible withsuch a cosmology. However, several alternative mecha-nisms (see, e.g., Ref. [34]) exist for generating a baryonasymmetry of the correct magnitude in an LTR context.Furthermore, since the maximum temperature Tmax at-tained by the thermal bath after inflation can be muchlarger than TRH—and indeed even above the weakscale—in LTR cosmologies, a variety of weak-scale baryo-genesis mechanisms (including electroweak baryogenesis)can still be viable even in scenarios with a reheatingtemperature TRH �OðMeVÞ. In fact, for certain baryogen-esis mechanisms, an LTR cosmology can even be benefi-cial [35], since the Universe expands more rapidly duringreheating, when the Universe is effectively matter-dominated, than during a radiation-dominated era. As aconsequence, sphaleron interactions fall out of equilibriumearlier in an LTR cosmology than in the standardcosmology.In our bulk-axion DDM model, the particular mecha-

nism responsible for generating the baryon asymmetry ofthe Universe has virtually no impact on the dark-matterphenomenology of the model. Consequently, since viablebaryogenesis mechanisms do exist for reheating tempera-tures TRH �OðMeVÞ, we will simply assume that somesuch mechanism produces a baryon asymmetry of theobserved magnitude in what follows.In summary, while stringent constraints exist on theories

with large extra dimensions, these constraints can be


055013-16

satisfied by adopting an LTR cosmology with 4 MeV &TRH & 20 MeV and a compactification manifold for whichthe astrophysical bounds listed above may consistently besatisfied for a given choice of MD and fX. Since �

tot isgenerated via nonthermal means in our bulk-axion model,as discussed in Sect. III, such a low value of TRH is not animpediment to obtaining a dark-matter relic abundance�

tot � �CDM. In fact, as shown in Ref. [2], adopting anLTR cosmology is actually an asset in terms of generatinga dark-matter relic abundance of the correct magnitude.

Likewise, since the relationship between fX and fX neednot be identical to the relationship between MP and MD,constraints which concern the effects of KK gravitons canbe satisfied without imposing equally severe restrictions on

the parameters fX, Mc, and �G which govern the proper-ties of the dark-matter ensemble. Indeed, the only signifi-cant model-independent constraint on these parametersturns out to be the constraint quoted in Eq. (4.1) from testsof Newton’s law at short distances.

B. Axion production with subsequent detection:Helioscopes and light shining through walls

We now address the constraints which relate directly tothe phenomenological, astrophysical, and cosmologicalimplications associated with the KK tower of axion fieldswhich constitute the DDM ensemble in our model. Webegin by discussing the limits derived from a wide varietyof experiments designed to detect axions and axionlikeparticles via their interactions with the photon field. (Forextensive reviews of these experiments, see Refs. [36,37].)To date, none of these experiments have seen any conclu-sive evidence for such particles, and the null results ofthese experiments therefore imply constraints on the effec-tive couplings between such axionlike particles and thephoton field.

In order to determine how the results of the experimentslisted above serve to constrain the parameter space of ourbulk-axion DDM model, it is useful to divide those experi-ments into several broad classes, based on the sort ofphysical process each probes. One important class of ex-periments comprises those in which axions are producedvia their interactions with the fields of the SM and thensubsequently detected via those same interactions. Theseinclude helioscope experiments such as CAST [38] and

‘‘light-shining-through-walls’’ (LSW) experiments such asBEV, GammaeV, and ALPS. Searches for coherent con-version of solar axions to X-ray photons in germanium andsodium-iodide crystals via Bragg diffraction which havebeen performed at experiments such as DAMA [39],TEXONO [40], SOLAX [41], and COSME [42] also fallinto this category. The characteristic which distinguishesexperiments in this class from others is that these experi-ments are affected by decoherence phenomena. Indeed, ithas been observed [3] that in theories with bulk axions,such phenomena result in a substantial suppression of therate for any process involving the production and subse-quent decay of axion modes relative to naıve expectations.Let us briefly review the origin of this suppression by

focusing on the interaction between the photon field andthe axion KK modes given in Eq. (2.5). (The results for thecoupling between these modes and the other SM fields arecompletely analogous.) We begin by defining a state

a0 � 1ffiffiffiffiN

p XNn

rnan; (4.5)

which represents the particular linear combination of KKeigenstates an that couples to any physics on the brane,such as F��

~F�� or any pair of SM fields. HereN � fX=Mc

denotes the number of modes in the sum. Written in termsof a0, the relevant term in the interaction Lagrangianbecomes

Lint 3�c�

ffiffiffiffiN

p

8�2fXa0F��

~F��: (4.6)

In other words, a0 couples to the SM fields with a strength

proportional toffiffiffiffiN

p=fX � 1=fX. Consequently, the cross

section for any physical process which involves axionproduction via interactions with the SM fields followedby subsequent detection via the same sorts of interactionswill take the form

ðtÞ / N2

f4X PðtÞ; (4.7)

where PðtÞ ¼ jha0ðtÞja0ðt0Þij2 is the probability for a statea0 created at time t0 to be in the same state a0 at time t. Itcan be shown that when N is large, PðtÞ is given by

PðtÞ ¼ 1

N2

�X�

~�8A4� þ 2

X�

X�0<�

~�4 ~�04A2�A

2�0 cos

�ð�2 � �02Þðt� t0Þ2p

��; (4.8)

where p is the initial momentum of the axion.At very early times, when t � t0, the cosine factor in

PðtÞ is approximately unity for all values of � and �0. Atsuch times, all of the terms in the sum appearing in thesecond term on the right side of Eq. (4.8) add coherently.

As a result, this term, combined together with the first term,yields a factor on the order of N2. However, as the systemevolves, the cosine terms will no longer sum coherently,and a random-walk behavior ensues, according to whichthe two terms combine to yield a factor of OðNÞ rather


055013-17

than of OðN2Þ. The time scale �D associated with thisdecoherence—or, more precisely, the scale at whichPðtÞ ¼ 0:1Pðt0Þ—is found to be [3]

�D � 10�5

�2p

m2PQ

�y2

N2� 1:32 10�29

�p

GeV

��fXGeV

��2s;

(4.9)

where y is defined in Eq. (2.12). Since �D is clearly quitesmall for any combination of p and fX values of experi-mental relevance, any method of detecting axions whichrelies on their production and subsequent detection willfeel the effect of this decoherence. By contrast, detectionmethods which rely on axion production without subse-quent detection (such as missing-energy signals at col-liders, energy dissipation from supernovae, etc.) or whichprobe for evidence of a cosmic population of relic axions(such as microwave-cavity experiments) will be unaffectedby this phenomenon.

The consequences of axion decoherence for physicalprocesses in the decoherence regime are readily apparent.In this regime, as discussed above, the term in brackets in

PðtÞ scales like N rather than N2; hence any cross sectionwhich takes the form given in Eq. (4.7) will scale withN � fX=Mc according to

ðt > �DÞ / N

f4X� 1

N

1

f4X: (4.10)

In other words, such cross sections are suppressed by anadditional factor of N relative to the naıve expectation

obtained by setting fX ! fX in Eq. (4.7). Thus, due tothe decoherence effect, any experimental bound on theeffective coupling Ga�� of a single four-dimensional axion

to the photon field which takes the form G2a�� < ðGmax

a��Þ2translates to a bound G2

a�� < ðGmaxa��Þ2=

ffiffiffiffiN

pfor a five-

dimensional axion, rather than to G2a�� < ðGmax

a��Þ2=N.

Given the parametrization for Ga�� given in Eq. (2.5), we

can phrase any such constraint as a bound on fX:

f X *c��

2�Gmaxa��

�Mc

fX

�1=4

: (4.11)

Using Eq. (2.7), we may rewrite this constraint in the form

f X *1

ð2�Þ13=10�c��

Gmaxa��

�6=5 1

M1=5c

¼ ð2:50 10�4Þc�ðGmaxa��Þ�6=5M�1=5

c : (4.12)

The most stringent limit from the class of experimentscategorized above (i.e., those for which the phenomenon ofdecoherence is relevant) is currently the Ga�� &8:8 10�11 GeV�1 bound obtained by CAST [38]. Themost stringent limit from crystalline detectors is theGa�� & 1:7 10�9 GeV�1 bound from DAMA [39], andlimits on Ga�� from LSW experiments are typicallyroughly three orders of magnitude higher than the CASTlimit. The corresponding bound on fX from Eq. (4.12) is

f X * ð2:92 108Þc6=5�

�Mc

GeV

��1=5GeV: (4.13)

Note that even forMc at the experimental lower limit givenin Eq. (4.1), the constraint in Eq. (4.13) is satisfied as longas fX * 5:58 1010 GeV.

C. Microwave-cavity experiments and direct detectionof dark-matter axions

Another class of experiments which place constraints onthe couplings of axions and axionlike fields to SM particlesconsists of those which involve the direct detection of acosmological population of axions. The most sensitiveexperiments in this class are those associated with dedi-cated microwave-cavity detectors such as ADMX [43] andCARRACK [44]. Detectors of this sort are used to searchfor the resonant conversion of dark-matter axions withmassma to photons with energies E� � ma in the presence

of a strong magnetic field. As a result, the observation of asignal at such a detector depends crucially on whether the

mass of the axion in question lies within the range ofphoton energies probed. The axion mass range currentlycovered by ADMX spans only from 1:9 10�15 GeV to3:5 10�15 GeV [43], and the projected future mass sen-sitivity extends only as high as 10�13 GeV. Likewise, theprojected sensitivity for CARRACK extends only as highas 3:5 10�14 GeV.As discussed in Ref. [2], the region of parameter space

which is the most interesting from a DDM perspective isthat within which y & 1 and mixing among the light axionKK modes is substantial, for it is this region within whichthe full tower contributes meaningfully to�

tot. Within thisregion of parameter space, the lightest mode in the towerhas a mass �0 � Mc=2. Taken in conjunction with thebound on Mc from modified-gravity experiments given inEq. (4.1), this result implies that �0 * 1:5 10�12 GeV inhighly mixed bulk-axion scenarios. The projected rangesfor both ADMX and CARRACK lie well below this thresh-old for �0. We therefore conclude that no meaningfulconstraints on bulk-axion DDM models can be derivedfrom the results of these experiments.

D. Axion production without subsequent detection:Stars and supernovae

We now turn to examine an additional class of con-straints on bulk-axion DDM models: those related to as-trophysical processes in which the a� are produced throughtheir interactions with the SM field, but never directlydetected. Among the constraints in this class are limits


055013-18

on axions, moduli, and other light scalars derived from thenonobservation of their would-be effects on the lifetimes,energy-loss rates, etc., of various astrophysical sourcessuch as stars and supernovae. These effects include thefollowing:

(i) Axions and other light fields whose interactions withthe particles of the SM are extremely weak andwhose mean free paths are consequently extremelylong can dissipate energy from stars extremely effi-ciently. Such dissipation can accelerate stellar cool-ing and result in observable alterations in stellar lifecycles, including the life cycle of our own sun.

(ii) Similarly, such light fields can carry away a sub-stantial fraction of the energy liberated by super-novae. Limits may therefore be placed on thestrengths of these interactions from the nonobserva-tion of such effects for supernova SN1987A.

(iii) A diffuse population of long-lived axions or KKgravitons initially produced by stars and superno-vae could decay at late times, distorting light-element abundances and producing an observableX-ray or �-ray signal in the keV–MeV range orhigher. No evidence for such a signal has been seenby EGRET, FERMI, HEAO, Chandra, COMPTEL,etc.

As is well known, these considerations lead to someof the most stringent constraints on standard, four-dimensional QCD axions. We now turn to examine howthese limits constrain the parameter space of generalizedbulk-axion models.

The primary distinction between processes in which thepresence of the a� is ascertained by direct detection andthose in which it is only inferred from an energy deficit isthat in the latter class of processes, the a� appear asparticles in the asymptotic final state. Thus, the contribu-tions from the individual a� to the overall event rate for anysuch process add not at the amplitude level, but at thecross-section level. The decoherence phenomena discussedin Sect. IVB are therefore irrelevant for such processes,

and the total cross section prodtot for the production of

‘‘missing energy’’ in the form of a� fields by any givenphysical process is simply the sum of the individual pro-

duction cross sections prod� for each axion species. Since

the effective coupling between each a� and any pair of SM

fields includes a factor ~�2A�=fX from mass mixing, asindicated in Eq. (2.17), each of these individual productioncross sections scales as

prod� / 1

f2Xð~�2A�Þ2: (4.14)

When it occurs, axion production will have a character-istic energy scale Ech determined by the surrounding envi-ronment. This energy scale may be associated, forexample, with the temperature of a star or supernovacore, or with the center-of-mass energy

ffiffiffis

pof a collider.

Provided that Ech � Mc (an assumption valid for all physi-cal contexts of relevance in bounding the large-extra-dimension scenarios considered here), it follows that � �Ech for a large number of a�. Such a� can be considered tobe effectively massless as far as production kinematics isconcerned, implying that to a very good approximation,

prod� depends on � exclusively through the coupling-

modification factor appearing in Eq. (4.14). (For thosemodes for which threshold effects are important, such an

approximation will overestimate prod� and result in an

overly conservative bound.) By contrast, prod� will be

effectively zero for those a� with masses � � Ech in anythermal environment due to Boltzmann suppression, andwill vanish outright in a nonthermal one. Therefore, it is

reasonable to evaluate prodtot by taking any additional fac-

tors in Eq. (4.14) to be essentially independent of � and bytruncating the sum over modes at �� Ech. Thus, we findthat

prodtot / @2ðEchÞ=f2X (4.15)

where the ‘‘effective’’ coupling @ðEchÞ is given by

@ðEchÞ �� XEch

�¼�0

ð~�2A�Þ2�1=2

: (4.16)

Since the number of modes contributing to prodtot is large by

assumption, and since their masses are closely spaced, it isgenerally legitimate to approximate @ðEchÞ by an integral

@ðEchÞ ��1

Mc

Z Ech

�0

ð~�2A�Þ2d��1=2

: (4.17)

The quantity @ðEchÞ clearly plays a crucial role in thephenomenology of bulk-axion scenarios. It is thereforeworth pausing a moment to examine in detail how @ðEchÞdepends on the physical scales fX,Mc, and�G. A straight-forward calculation shows that @ðEchÞ has the parametricscaling behaviors

@ðEchÞ �

8>><>>:

E3=2ch

M1=2c f2X

�4G

fX � �2G

McEch

Mc

1=2

fX � �2G

Mc:

(4.18)

The first case in Eq. (4.18) corresponds to y � 1, signaling

a highly mixed axion KK tower for which ~�2A� � ~�. Bycontrast, the second case corresponds to y � 1, signaling a

relatively unmixed axion KK tower for which ~�2A� �constant. These results for @ðEchÞ are illustrated in theleft panel of Fig. 4 for Ech ¼ 30 MeV, a value which isphysically meaningful in that it corresponds roughly to thecore temperature of supernova SN1987A. Remarkably, weobserve that @ðEchÞ experiences a suppression for y � 1.In other words, mixing within the axion KK tower acts tosuppress the magnitude of the total production cross sec-tion for processes in which the a� appear as missing


055013-19

energy. This is an important result, for it indicates thatconstraints on the parameter space of our bulk-axion DDMmodel derived from limits on axion production in stars,supernovae, colliders, etc., will be considerably weakerthan one might expect from naıve dimensional analysis.Moreover, this result applies more generally to any theoryinvolving KK towers of scalar fields whose squared-massmatrix contains both brane-mass and KK-mass terms.

In order to illustrate more explicitly the physical con-sequences of @ðEchÞ in our bulk-axion DDM model, welikewise display the behavior of the overall scaling factor

@2ðEchÞ=f2X for prodtot in the right panel of Fig. 4. The results

shown in this panel further illustrate a significant generalproperty of this scaling factor: namely, that the cross

section is actually suppressed not only for large fX, but

also for small fX, due to the parametric behavior of @ðEchÞdescribed in Eq. (4.18). Thus, for any given choice of �G

and Mc, there exists a maximum possible value for prodtot ,

which is only attained at some particular value of fX. Theseresults again illustrate the dramatic effect that @ðEchÞ canhave in suppressing prod

tot in our bulk-axion DDM model.Within the class of constraints from processes in which

axions are produced but not subsequently detected, use of@ðEchÞ allows us to translate experimental bounds on four-dimensional axion models into bounds on theories includ-ing towers of bulk scalars. The leading such bound isobtained from energy-loss limits from SN1987A. For astandard four-dimensional QCD axion, this bound isroughly [45]

fa * 4 108 GeV: (4.19)

By contrast, in the bulk-axion scenario under considerationhere, each a� light enough to be produced within thethermal environment of SN1987A can contribute to theoverall energy-dissipation rate. Since the temperature TSN

associated with the supernova core is roughly 30 MeV, theappropriate modification of Eq. (4.19) for a general axionwhich couples to hadrons with a coupling coefficient com-parable in magnitude to that of a QCD axion is

f X * ð4 108 GeVÞ@ðTSNÞ: (4.20)

It then follows that in highly mixed scenarios, this con-straint can be significantly weaker than the correspondingconstraint on KK-graviton production derived in Ref. [7],due to suppression by @ðTSNÞ. Indeed, the correspondingconstraint on KK-graviton production is directly ob-

tained by replacing fX ! MP and @ðTSNÞ ! TSN=Mc inEq. (4.20).While the SN1987A bound is indeed one of the most

stringent constraints on the QCD axion, it is not necessarilyapplicable for all general axions. This is because the boundquoted in Eq. (4.19) is predicated on the assumption thatnucleon bremsstrahlung (N þ N ! N þ N þ a) and otherhadronic processes dominate the rate for the production ofthe light scalar in question in the supernova core. Thispresupposes that the light scalar couples to nuclei with astrength comparable to that of a QCD axion. If this is notthe case, however, the constraints obtained from SN1987A

Mc 10 11 GeV

G 1 GeVG 10 GeVG 100 GeVG 1 TeVG 10 TeVG 100 TeV

106 108 1010 1012 1014 101610 15

10 12

10 9

10 6

0.001

1

1000

Log10 f X GeV

Log

10T

SNT

SNM

c1

2

G 1 TeV

Mc 10 13 GeVMc 10 11 GeVMc 10 9 GeVMc 10 7 GeV

106 108 1010 1012 1014 101610 15

10 13

10 11

10 9

10 7

Log10 f X GeV

Log

102

TSN

f X2G

eV2

FIG. 4 (color online). The dimensionless ‘‘effective coupling’’ factor @ðEchÞ defined in Eq. (4.16), shown as a function of the relevantscalesMc, fX, and�G. In the left panel, we display curves of @ðEchÞ, each corresponding to a particular value of�G and normalized tothe value ðEch=McÞ1=2 taken by @ðEchÞ in the absence of mixing, as a function of fX with fixedMc ¼ 10�11 GeV. It is readily apparentthat the net effect of mixing within the KK axion tower is to significantly suppress this effective coupling, thereby loosening thecorresponding production-cross-section constraints. In the right panel, we display curves showing the overall cross-section-suppression factor @2ðEchÞ=f2X as a function of fX for fixed � ¼ 1 TeV, each corresponding to a particular value of Mc. For eachset of curves, we have taken � ¼ gG ¼ 1, and have chosen Ech ¼ 30 MeV, which corresponds roughly to the core temperature TSN ofSN1987A.


055013-20

energy-loss limits can differ considerably from the stan-dard QCD-axion bound. For example, if from among theSM particles, the general axion couples only to the photonfield, the dominant production processes will be e�� !ea�, p

þ� ! pþa�, and pþn ! pþn�a�. In this case, theconsiderably weaker bound [46]

f X * ð2:32 106 GeVÞc� (4.21)

is obtained for a four-dimensional field. Translating thisresult to the case of a KK tower of axions, as above, we findthat

f X * ð2:32 106 GeVÞc�@ðTSNÞ: (4.22)

Furthermore, in general axion models, c� may not neces-

sarily be of Oð1Þ. In other words, the SN1987A constraintis sensitive to the Uð1ÞX and SUð2Þ Uð1ÞY charges of thefields in the model, and is thus highly model dependent.

An analogous limit on fX can be derived from observa-tions of the lifetimes of globular-cluster (GC) stars. Theambient temperatures TGC of such objects are onlyOð10 keVÞ, so axion production primarily proceedsthrough the Primakoff processes �þ e� ! aþ e� and�þ nZ ! aþ nZ, where nZ denotes a nucleus withatomic number Z. (Note that the dominant processes inthis environment differ from the axion-nucleon-nucleonbremsstrahlung processes which dominate the axion-production rate in supernovae.) Such a bound will thereforearise for any general axion for which c� � 1, regardless of

whether or not it couples to the gluon field. The observa-tion limit on axion production in GC stars is commonlyphrased as an upper bound on the effective coupling Ga��

between a standard, four-dimensional axion (or any othersimilar particle) and a pair of photons, and the currentbound is Ga�� & 1 10�10 GeV�1 [47]. Since TGC �10 keV, the corresponding bound on fX is

f X * ð1:16 107 GeVÞc�@ðTGCÞ: (4.23)

Note that this constraint is independent of the SN1987Abounds, as it differs from the latter in two significant ways.First, because the relevant production process involves thecoupling of the axion modes to photons rather than tonuclei, it depends on c� alone and not on cg. Second, since

TGC � TSN, far fewer of the a� will be produced with anysignificant frequency in GC stars. Consequently, the en-hancement factor from the sum over kinetically accessibleaxion modes for GC stars is far smaller.

Finally, bounds similar to those from SN1987A and GCstars can also be derived from the nonobservation of effectsrelated to axion production in other astrophysical sources,such as our own sun [48]. However, these bounds are foundto be subleading in comparison with the SN1987A andGC-star constraints, essentially because they take place infar cooler environments, where the number of kinemati-

cally accessible modes is even further suppressed by thecutoff at Ech inherent in @ðEchÞ.

E. Axion production at colliders

We now consider the collider constraints applicable toour bulk-axion DDM model. Due to the highly suppressedcouplings between the axion and the SM fields in standardfour-dimensional axion models, collider data have virtu-ally no relevance in constraining the parameter space ofsuch models. Nevertheless, because of the huge multiplic-ity of light modes that arises in theories with light bulkfields in large extra dimensions, the net contribution to theevent rates for certain processes from all of these modestaken together can potentially yield observable signals. Forexample, modes which are stable on collider time scales allappear as missing energy, and can lead to signals in chan-nels such as pp ! jþ 6ET and pp ! �þ 6ET . In addition,the heavier, more unstable modes which decay beforeexiting the detector can potentially give rise to additionalsignature patterns which may include displaced vertices.Indeed, we have already discussed in Sect. IVA howcurrent limits from LHC data constrain the parameterspace of one such bulk field—the higher-dimensionalgraviton—for which the monojet and monophoton chan-nels mentioned above are of particular importance. Sincethe a� in our bulk-axion model couple to the fields of theSM in much the same manner as KK gravitons, it is nosurprise that the collider phenomenology of the a� turnsout to be quite similar to that of KK gravitons.We begin by discussing those signals which arise due to

the combined effect of the a� which are sufficiently long-lived so as to manifest themselves in a collider detector asmissing energy. Collider processes in which the a� appearas 6ET are yet further examples of the class of processesdiscussed in the previous section in which axions areproduced but not subsequently detected. The net crosssection for any such process is therefore likewise sup-pressed by axion mixing in the manner described inEq. (4.15), with Ech given by the center-of-mass energyffiffiffis

pof the collider.Which specific channels are relevant for the discovery of

a bulk axion at hadron colliders depends crucially on howthe five-dimensional axion couples to the SM fields, and inparticular on whether or not it couples appreciably to eitherlight quarks or gluons. For a field with an Oð1Þ value ofeither cg or cq (where q ¼ fu; d; s; cg), the principal dis-

covery channel at both the Tevatron and the LHC is pp !jþ 6ET , a channel which is also one of the principal dis-covery channels for KK gravitons. Thus, in order to obtaina rough estimate of the constraints on the parameter spaceof our bulk-axion model from the null results of monojetsearches, we translate the bound on the fundamental scale

MD established by such searches into a bound on fX. Thecross section for KK-graviton production in associationwith a single jet at a hadron collider in a theory with n


055013-21

large, flat extra dimensions of equal length compactified onan n-torus, including contributions from all kinematicallyaccessible modes, is roughly proportional to [7]

prodðpp ! jþGÞ /� ffiffiffi

sp2�

�n 1

Mnþ2D

: (4.24)

This implies that a bound of the form MD >MminD can be

translated into a rough bound on the parameter space of ourbulk-axion model of the form

@2ð ffiffiffis

p Þf2X

&

� ffiffiffis

p2�

�n 1

ðMminD Þnþ2

; (4.25)

where @ðEchÞ is defined in Eq. (4.16). While this approxi-mate bound does not take into account the differences incoupling structure between KK graviton and axion fields orthe sum over polarizations for a massive graviton, it issufficient to obtain parametric estimates of the resulting

constraints on our three fundamental parameters fX, Mc,and �G.

In Fig. 5, we indicate the rough bounds on the parameterspace of our bulk-axion DDM model which can bederived in this manner, given a chosen value of Mmin

D .The contours shown in this figure correspond to constraintsof the form MD >Mmin

D for the illustrative values MminD ¼

f1; 10; 100g TeV.We now compare these results to actual constraints on

MD from current experimental data and examine the pro-jected LHC reach for our bulk-axion DDM model. Themost stringent constraints from LHC data (which indeedcome from the pp ! jþ 6ET channel) were given inSect. IVA. Estimates of the future LHC reach for a theorywith a single extra dimension are Mmin

D � f14; 17g TeV atintegrated luminositiesLint ¼ f10; 100g fb�1, respectively[49]. Likewise, Tevatron data imply a limit Mmin

D �

2:4 TeV for a theory with a single extra dimension [49].Comparing these results to those in Fig. 5, we see thatcurrent collider constraints, while quite stringent, do notsignificantly impact the preferred region of parameterspace for our bulk-axion DDM model, even in cases inwhich the axion couples to one or more strongly interactingSM fields with anOð1Þ coupling coefficient. In such cases,since the most stringent current LHC limits imply a boundof roughly Mmin

D � 1 TeV, the region of the parameter

space of our model excluded by these limits roughly cor-responds to the green-shaded regions shown in Fig. 5.Since the green exclusion regions in this figure embodythe most stringent such limits applicable to our DDMmodel, we shall take these to represent our collider con-straints throughout the rest of this paper. However, we notethat for photonic axions and other axion species which donot couple directly to quarks or gluons, the correspondingcollider constraints (which arise from channels such aspp ! �þ 6ET) are somewhat weaker.Before concluding this section, there is one important

point which deserves emphasis. The collider processes wehave been discussing thus far are those whose event ratesreceive their contributions from the low-lying modes in thetower—i.e., those a� with lifetimes �� * 10�12 s. By con-trast, those heavy a� with lifetimes �� & 10�12 s tend todecay to pairs of SM fields within the detector volume. Thedecays of such states can in principle give rise to anentirely different set of signature patterns. For example, apromptly decaying a� which couples to light quarks orgluons as well as photons would in principle contribute toevent rates in the pp ! jjj and pp ! ��þ j channels.However, since the total event rate in these channels re-ceives contributions from a broad spectrum of a� withdifferent �, many event-selection techniques which areparticularly useful in standard searches for new physics

FIG. 5 (color online). Excluded regions of the ðfX;McÞ parameter space of our DDM model in which the collider constraint inEq. (4.25) is violated forMmin

D ¼ 1 TeV (green);MminD ¼ 10 TeV (green and blue); andMmin

D ¼ 100 TeV (green, blue, and purple). As�G increases, we see that satisfying the collider constraints becomes increasingly easy, particularly for small fX. In each case, we havetaken � ¼ gc ¼ 1 and assumed that the axion couples to at least one light, strongly interacting SM particle with an Oð1Þ couplingcoefficient cg or cq.


055013-22

in these channels cannot be applied to a tower of decayingbulk axions. For example, since the set of decaying axionscannot be characterized by a single, well-defined mass orcross section, no identifiable peak can be expected toappear in the invariant-mass distribution for the decayproducts of the heavy axions. Such considerations renderthe results of standard searches for new physics in thesechannels inapplicable to our bulk-axion model—and in-deed to DDMmodels in general. Moreover, they also likelyrender the identification of a conclusive signal of non-standard dark-matter physics in these channels particularlychallenging. Nevertheless, the information that could po-tentially be revealed about the nature of the dark sector viasuch an identification is of sufficient magnitude and im-portance that an analysis of the discovery potential in thesechannels is an interesting topic for future study.

F. Axion decays and distortions of the cosmicmicrowave background spectrum

Up to this point, we have considered those phenomeno-logical constraints on our DDMmodel which are related tothe production of particles which compose our bulk-axionensemble, both with and without their subsequent detec-tion. By contrast, we now turn to discuss an entirely differ-ent set of phenomenological constraints, namely thosewhich arise due to the potential decays of a preexistingcosmological population of such particles. Indeed, suchconstraints emerge generically in all dark-matter scenariosin which the dark sector contains unstable, long-livedparticles, and can be derived from observational limits onthe physical consequences of the late decays of thoseparticles.

There are many considerations which can be used toplace such limits on scenarios involving decaying dark-matter particles. For example, photons produced via the

decays of such particles can yield observable distortions inthe CMB spectrum; contribute to the diffuse extragalacticX-ray and gamma-ray backgrounds; upset BBN predic-tions for the primordial abundances of light elements;and result in unacceptably large entropy production duringcritical epochs in the history of the Universe. Constraintson dark-matter candidates from considerations of this sortdepend not only on the decay rate of the particle species inquestion, but also on the relic abundance of that species.For this reason, the constraints applicable to single-particlemodels of dark matter are generally not directly applicableto models within the DDM framework. It is thereforenecessary to revisit the observational limits on dark-matterdecays within the context of our bulk-axion model ofdynamical dark matter and assess how these limits con-strain the parameter space of this model.In this section, we begin our analysis of the con-

straints on the late decays of the a� in our bulk-axionDDM model by examining observational limits on thedistortions of the CMB which such decays can induce.The type of CMB distortion to which a late-decayingparticle contributes depends on the time at which thatparticle decays. In the very early Universe, photons pro-duced by particle decays are brought into thermal andkinetic equilibrium with CMB photons via a number ofprocesses. The dominant processes by which newlyproduced photons can equilibrate thermally with CMBphotons are double-Compton scattering (e�� ! e��)and bremsstrahlung (e�X� ! e�X��, where X� isan ion). However, once these processes freeze out,photons produced from a� decays are unable to thermallyequilibrate with the radiation bath, resulting in the genera-tion of a nonzero value for the pseudodegeneracy parame-ter �. The interaction rates for these processes are givenby [50]

�DC � 5:73 10�39

�1� Yp

2

�ð�Bh

2Þ�Tnow

2:7 K

�3=2

�tMRE

t

�9=4

GeV

�BR � 1:57 10�36

�1� Yp

2

�ð�Bh

2Þ3=2�Tnow

2:7 K

��5=4�tMRE

t

�13=8

GeV;

(4.26)

where Tnow � 2:725 K is the present-day CMB tempera-ture, Yp � 0:23 is the helium mass fraction,�B � 0:044 isthe baryon density of the Universe, and h � 0:72 is theHubble constant. (Note that since z is quite large during theentirety of the relevant time frame, we have here approxi-mated 1þ z � z.) Once these processes freeze out, in thesense that the rates given in Eq. (4.26) drop below theexpansion rate H of the Universe, photons produced bya� decay will no longer be able to attain thermal equilib-rium with the CMB photons. Even after double-Comptonscattering and bremsstrahlung effectively shut off, a num-ber of photon-number-conserving interactions still serve tobring photons produced at even later times into kinetic

equilibrium with the radiation bath. Dominant among theseprocesses is elastic Compton scattering (e�� ! e��),which efficiently serves to bring photons produced by a�decays into kinetic equilibrium until a much later timetEC � 9 109 s, at which point this process too effectivelyfreezes out. However, since elastic Compton scatteringconserves photon number, it cannot similarly suffice tobring those photons into thermal equilibrium. As a result,CMB distortions in the form of a nonzero value for thepseudodegeneracy parameter � can be generated by a�decays during this epoch. In addition, after elasticCompton scattering freezes out, photons produced by a�decay achieve neither kinetic nor thermal equilibrium with


055013-23

the radiation bath. As a result, these photons no longercontribute the generation of�, but instead contribute to thegeneration of a Compton y parameter (here denoted yC, soas to distinguish it from the ratio y ¼ Mc=mX). Finally, att� 1013 s, matter and radiation decouple, and any a�decays occurring after this point do not affect the CMB,but instead simply persist as a contribution to the diffusephoton background. This last sort of contribution will bedealt with separately, in Sect. IVG.

We thus see that axion decays have the potential togenerate both a nonzero � and a nonzero yC. We cantherefore establish constraints on our bulk-axion DDMmodel by calculating the theoretical predictions for thesequantities in our model and comparing these predictions toobservational data.

We begin our analysis of CMB distortions from a�decays by addressing those decays which result in thegeneration of the pseudodegeneracy parameter �. In gen-eral, provided that the additional contribution �� to the

photon energy density �� from the decay of the a� fields is

small compared to the total ��, the time-evolution of� can

be described by the equation [50,51]

d�

dt¼ d�a

dt��ð�DC þ �BRÞ: (4.27)

Here �DC and �BR are the interaction rates for double-Compton scattering and bremsstrahlung, respectively, andd�a=dt denotes the differential contribution to � fromaxion decay. For an arbitrary d�a=dt, the solution to thisdifferential equation takes the form

�ðtÞ ¼ exp

�4

5ð2CBRt

�5=8 þ CDCt�5=4Þ

�Z t

te

�d�a

dtðt0Þ

�exp

�� 4

5ð2CBRt

0�5=8 þ CDCt0�5=4Þ

�dt0; (4.28)

where te � 1:69 103 s is the time scale associated withelectron-positron annihilation in the early Universe, andwhere the quantities CDC and CBR are constants related tothe double-Compton-scattering and bremsstrahlung rates�DC and �BR in Eq. (4.26) by �DC � CDCt

�9=4 and �BR �CBRt

�13=8. Moreover, the differential contribution d�a=dtto� from axion decays is given by the standard expressionfor contributions due to the late injection of photons from ageneric source:

d�a

dt¼ 1

2:143

�3

��

d��

dt� 4

n�

dn�dt

�: (4.29)

In general, the rate of change in the photon energydensity is given by the Boltzmann equation for the evolu-

tion of ��. In our bulk-axion DDM model, this equation

includes a source term from each decaying state in thedark-matter ensemble. Thus, at late times, after all of thea� have already begun oscillating coherently and the con-tribution to �� from inflaton decays can safely be ne-

glected, we find that

d��

dt¼ �4H�� þX

�

BRð2�Þ� ��; (4.30)

where BRð2�Þ� is the branching fraction of a� into a pair of

photons. Note that the source term in the Boltzmann equa-tion for �� is simply a sum of the contributions from the

various a� fields. Using Eq. (4.30), along with the relations

1

ðR4��ÞdðR4��Þ

dt¼ 1

��

�d��

dtþ 4H��

�;

1

ðR3n�ÞdðR3n�Þ

dt¼ 1

n�

�dn�dt

þ 3Hn�

�; (4.31)

we can rewrite Eq. (4.27) in the form

d�a

dt¼ 1

2:143

�3

��

X�

BRð2�Þ� �� 8

n�

X�

BRð2�Þ� ��

��

�

�: (4.32)

For the purpose of establishing a conservative bound, we focus here on the case of a purely photonic axion. As we saw inSect. III B, the contribution to �� from intraensemble decays is negligible for any a� which decays on time scales relevantfor the generation of CMB distortions. It is therefore justifiable to approximate �� by the expression for �ða ! ��Þ givenin Eq. (3.15) and thus to take BRð2�Þ

� � 1. Since the energy density �� associated with each a� is given in Eq. (3.7), we findthat in this approximation, the first source term on the right side of Eq. (4.32) takes the form

X�

BRð2�Þ� �� 1

22G�m

4X

X�

�

�t2�

t1=2RH

�ð~�2A�Þ4e

�G��3

f2X

ð~�2A�Þ2ðt�tGÞ 8><>:t1=2RH t

�2 t & tRHt�3=2 tRH & t & tMRE

t1=2MREt�2 t * tMRE;

(4.33)


055013-24

where G� is defined below Eq. (3.15). The second termtakes the same form, but with one factor of � fewer in thesummand.

In principle, one could evaluate this sum numerically ateach moment in time, and then use these results to numeri-cally solve Eq. (4.32). However, we find that by making afew additional well-motivated approximations, we canobtain a closed-form, analytical result for d�a=dt. Webegin by dividing the tower into sections, based on thetwo criteria which determine the dependence of �� and ��

on �. The first of these is whether the oscillation-onset timefor a given a� is within the staggered regime (i.e., t� > tG),or the simultaneous turn-on regime (i.e., t� ¼ tG). In theformer case, t� depends on � according to Eq. (3.6); in thelatter case, t� is independent of �. The second pertinentcriterion concerns the relationship between � and thequantity

�trans � �m2X=Mc; (4.34)

introduced in Ref. [2]. This quantity corresponds roughlyto the transition point between the small-� regime, inwhich the a� are highly mixed, and the large-� regime,in which mixing is negligible. Indeed, for � � �trans, we

find that ~�2A� � ffiffiffi2

p~�=ð1þ �2=y2Þ1=2, while for � �

�trans, we find that ~�2A� � ffiffiffi2

p. Given these criteria, our

first approximation will be to replace ~�2A� with its asymp-totic large-� form for all � > �trans, and with its asymptoticsmall-� form for all � < �trans. Our second will be toapproximate the sum over � by a set of source-term inte-grals Iiðm; n; �; �; �min; �maxÞ, each corresponding to adifferent regime in the tower of modes characterized by aparticular dependence of the integrand on �. These source-term integrals may be evaluated analytically by making useof the identity

Iiðm; n; �; �; �min; �maxÞ � �Z �max

�min

�me��nd�

¼ �1

n��ðmþ1Þ=n

��

�mþ 1

n; ��n

min

��

�mþ 1

n; ��n

max

��; (4.35)

which is valid for n > 0 and any real values of m, �, and �. Here �ðs; xÞ denotes the incomplete gamma function:

�ðs; xÞ �Z 1

xts�1e�tdt: (4.36)

Employing the approximations discussed above, we find that the first source term on the right side of Eq. (4.32)reduces to

X�

BRð2�Þ� �� ¼ 2G�

2

Mc

X4i¼1

Iiðmi; ni; �i; �i; �CMBi�1 ; �CMB

i Þ 8><>:t1=2RH t

�2 t & tRHt�3=2 tRH & t & tMRE

t1=2MREt�2 t * tMRE:

(4.37)

Inserting this result (and the analogous result for the second source term) into Eq. (4.29) and using the fact that �-typedistortions are generated by decays occurring within the RD era, we obtain the result

d�a

dt� 0:935 G�

2

Mct3=2

�3

�eq�

X4i¼1


i Þ � 8

neq�

X4i¼1

Iiðmi � 1; ni; �i; �i; �CMBi�1 ; �CMB

i Þ�; (4.38)

where the expressions for �i, �i, mi, and ni valid in eacha� regime are listed in Table I. Note that in obtaining thisexpression, we have assumed that the additional contribu-tions to n� and �� due to the injection of photons from a�are sufficiently small that these quantities can be approxi-

mated by the equilibrium expressions neq� ¼ 2�ð3ÞT3=�2

and �eq� ¼ �2T4=15. Furthermore, we have used the fact

that the time frame during which CMB distortions to� canarise lies entirely within the RD era. Obtaining a final resultfor the magnitude of �-type distortions to the CMB

TABLE I. Values ofmi, ni, �i, and �i which correspond to different regimes, labeled by the index i, in a generic axion tower, for usein Eqs. (4.38) and (4.49). The symbol G denotes the specific value of , as defined in Eq. (3.3), which corresponds to tG.

i Oscillation regime Mixing regime mi ni �i �i

1t� > tG

� < �trans 3 5 4t�1=2RH ½1þ �2=y2��2 2G�tðfXmXÞ�2½1þ �2=y2��1

2 � � �trans �1 3 4m4Xt

�1=2RH 2G�tf

�2X

3t� ¼ tG

� < �trans 5 5 tG

G t3=2�G

RH ½1þ �2=y2��2 2G�tðfXmXÞ�2½1þ �2=y2��1

4 � � �trans 1 3 m4Xt

G

G t3=2�G

RH 2G�tf�2X


055013-25

engendered by the presence of a tower of decaying DDMaxions is then simply a matter of substituting the result ford�a=dt in Eq. (4.38) into Eq. (4.28) and numericallyevaluating the integral for a given choice of inputparameters.

The contribution to yC from the late decays of the a�may be evaluated in much the same way as the correspond-ing contribution to �. The decays which contribute to yCare those which occur during the window 9 109 & t &1:2 1013 s, during which the rate �EC �H associated

with elastic Compton scattering can no longer bring thephotons from a� decay into kinetic equilibrium eventhough radiation has yet to decouple from matter. Theevolution of yC is governed by the relation [52]

dyCdt

¼ 1

4��

d��

dt: (4.39)

Proceeding with the mode sum as above and adopting thesame approximations as above, we find that

dyCdt

� 2G�2

Mc�eq�

X4i¼1


i Þ 8><>:t�3=2 t & tMRE

t1=2MREt�2 tMRE & t & tLS

0 t * tLS;

(4.40)

where tLS � 1:19 1013 s is the time of last scattering andIiðm; n; �;�; �min; �maxÞ is once again given by Eq. (4.35).Note that since matter-radiation equality occurs prior tolast scattering, at around tMRE � 1011 s, the epoch duringwhich a� decays can affect yC straddles both the RDand MD eras. Numerically evaluating the expression inEq. (4.40) from tEC to tLS, we obtain our final results for yCdistortions due to late a� decay.

In order to assess the CMB constraints on the parameterspace of our bulk-axion DDMmodel, we now compare theresults obtained by numerically integrating Eqs. (4.28) and(4.40) with observational limits on � and yC. The currentlimits on these quantities are [47]

j�j< 9 10�5; yC < 1:2 10�5: (4.41)

The bound on yC for a photonic axion with c� ¼ 1 yields

the constraints on fX, Mc, and �G shown in Fig. 6. In this

figure, we display contours of the values of yC in ðfX;McÞspace which arise in a bulk-axion model with�G ¼ 1 GeV(left panel) and with �G ¼ 1 TeV (right panel). In eachcase, we have taken � ¼ gG ¼ ¼ 1, HI ¼ 1 GeV, andTRH ¼ 5 MeV. Contours indicating y � Mc=mX ¼ 1(solid red line) and y ¼ f0:01; 0:1; 10; 100g (dashed redlines) have also been superimposed. For each panel, it isevident that our bulk-axion DDMmodel amply satisfies the

CMB constraints for all relevant values of fX and Mc,regardless of the value of y.It turns out that the constraints from the corresponding

bound on � in Eq. (4.41) are even less stringent than those

G 1 GeV

yC

1010

yC

1015

yC

1020

yC

1025

yC

1030

yC

1035

6 8 10 12 14 1612

11

10

9

8

7

6

f X GeV

Mc

GeV

G 1 TeV

yC

1015

yC

1020

yC

1025

yC

1030

yC

1035

yC

1020

yC

1025

yC

1030

6 8 10 12 14 1612

11

10

9

8

7

6

f X GeV

Mc

GeV

FIG. 6 (color online). Contours of the CMB Compton-y-parameter distortion yC (black lines) produced as a result of axion decays ina bulk-axion DDM model with �G ¼ 1 GeV (left panel) and �G ¼ 1 TeV (right panel). In each case, we have assumed a photonicaxion with c� ¼ 1 and have taken � ¼ gG ¼ ¼ 1, with HI ¼ 1 GeV and TRH ¼ 5 MeV. Contours corresponding to y � Mc=mX ¼1 (solid red line) and to y ¼ f0:01; 0:1; 10; 100g (dashed red lines) are also shown. For each panel, it is evident that our bulk-axionDDM model amply satisfies the CMB constraints in Eq. (4.41) for all relevant values of fX and Mc, regardless of the value of y.


055013-26

from the bound on yC. Thus, we conclude that both theyC-type and �-type distortions which result from a� de-cays in our bulk-axion DDM model are well below presentexperimental sensitivities. Indeed, no meaningful con-straint arises for our bulk-axion DDM model from presentlimits on distortions in the CMB.

As we have discussed, neither � nor yC can be affectedby any photons which are produced by a� decays at timest * tLS, after radiation and matter decouple. Such photonsdo, however, contribute to the diffuse photon background.In the next section, we will discuss the physical effects ofthis diffuse photon background in detail.

G. Axion decays and contributions to the diffuse X-rayand gamma-ray backgrounds

As mentioned above, the potentially observable effectsof late photoproduction from axion decays include notonly distortions of the CMB, but also imprints on thediffuse X-ray and gamma-ray backgrounds. Obser-vational limits on such imprints from instruments such asHEAO [53], COMPTEL [54], XMM, and Chandra [55]therefore impose additional constraints on the parameterspace of our bulk-axion DDM model. As discussed inSect. III, there are two cosmological populations of decay-ing a� whose decays to photons can potentially leaveobservable imprints on the diffuse X-ray and gamma-raybackgrounds. The first is the population of cold axionsproduced by vacuum misalignment, which collectivelycompose the DDM ensemble. The second is the far smallerpopulation of axions produced by interactions among theSM fields in the thermal bath after inflation. While theformer population provides a far greater contribution to�tot, the latter population contains a far larger proportionof heavier, more unstable a�, as indicated in Fig. 3. It is notclear a priori which population yields the more stringentconstraint. Thus, it is necessary to examine the contributionto the diffuse photon background from each of these pop-ulations in turn.

A photon produced at time t with initial energy E�ðtÞwill only contribute to the diffuse photon background if theUniverse remains transparent to electromagnetic radiationover the entire range of energies through which that photonredshifts as the Universe evolves from t to tnow. A detailedanalysis of the time scales and photon-energy ranges forwhich this transparency condition is attained is presentedin Ref. [56]. Roughly speaking, the transparency windowspans an energy range 1 keV & E� & 10 TeV and a time

range 1012–1014 s & t & tnow, with the lower limit depend-ing on the particular value of E�. Motivated by these

results, we approximate the Universe to be transparent toall photons with energies which fall within this range at alltimes t > tLS and opaque to all photons otherwise. Thisapproximation yields a conservative bound. Moreover, weemphasize that since the dominant contribution to thediffuse X-ray and gamma-ray flux in our model is due to

modes which decay at much later times t � tLS, our resultsare essentially insensitive to the precise contours chosenfor the transparency window.The calculation of the photon flux due to late a� decays

proceeds in a manner similar to the calculation of the fluxfrom KK-graviton decays outlined in Ref. [31]. TheBoltzmann equation for the number density n� of photons

in the presence of a tower of decaying a� takes the form

_n � þ 3Hn� ¼ 2X�


��

�; (4.42)

where once again �� is given by Eq. (3.8). Solving thisequation for n� as a function of time, we obtain

n�ðtÞ ¼ 2sðtÞsLS

X�

BRð2�Þ�

��ðtLSÞ�

½1� e��ðt�tLSÞ�; (4.43)

where sðtÞ is the entropy density of the Universe at time t,and sLS is the entropy density of the Universe at the time oflast scattering. The present-day differential energy spec-trum dn�=dE� of these photons may readily be computed

from the relation

dn�dE�

¼ dn�dt

dt

dz

dz

dE�

; (4.44)

where z is the cosmological redshift and E� is the photon

energy at redshift z. The first of these factors may beobtained by explicitly differentiating Eq. (4.43) with fixeds ¼ snow, which yields a series of terms of the form�

dn�dt

��¼ 2

�snowsLS

�BRð2�Þ

� ��

��ðtLSÞ�

e��ðtnow�tLSÞ;

(4.45)

one for each different value of �. The second factor inEq. (4.44) may be obtained by noting that the relationshipbetween time and redshift during the present, matter-

dominated era is well approximated by t ¼ tnowð1þzÞ�3=2. Consequently, for each value of � we have�

dt

dz

��¼ � 3

2tnow

�2E�

�

�5=2

(4.46)

during the epoch of interest. The third factor in Eq. (4.44)may be obtained by noting that each of the photons pro-duced by an axion tower state a� which decays at redshift zwill be monochromatic, with energy �=2, at the moment ofdecay. This implies that the present-day energies of suchphotons are given by E�ð1þ zÞ ¼ �=2, and hence that for

each value of �, we have�dz

dE�

��¼ � �

2E2�

: (4.47)

Combining these expressions and summing over �, wearrive at a general formula for the contribution to thediffuse photon flux produced by the tower of decaying a�:


055013-27

dn�dE�

��now¼ 6tnow

ffiffiffiffiffiffiffiffiffi2E�

q �snowsLS

�X�


��ðtLSÞ�5=2

e��ðtnow�tLSÞ: (4.48)

Calculating the contribution to the diffuse X-ray and gamma-ray backgrounds in our bulk-axion DDM model is thensimply a matter of applying Eq. (4.48) to the contribution from the two relevant populations of decaying axions discussedabove.

We begin by addressing the contribution from the population of axions produced by vacuum misalignment—i.e., theDDM ensemble itself. Once again, we focus our attention on the case of a photonic axion, for which �� ða ! ��Þ andBRð2�Þ

� � 1. In this case, we find that the contribution to the present-day diffuse photon background from the collective

decays of the a� fields is given by

dn�dE�

��now¼ 3


qG�

2m4X

�snowsLS

�X�

�t2�t

1=2MRE

t2LSt1=2RH

��3=2ð~�2A�Þ4e

G��3

f2G

ð~�2A�Þ2ðtnow�tGÞ: (4.49)

Just as for the contributions to� and yC in Sect. IV F, we approximate the sum over axion modes appearing in Eq. (4.49) asan integral over �. The lower limit of integration is determined by the requirement that in order for a photon with redshiftedenergy E� to have been produced by the decay of the axion species a� before present day, we must have � � 2E�.Likewise, photons which decay before the processes which equilibrate them with the radiation bath freeze out will notcontribute to features in the diffuse photon background. Thus, the upper limit of integration is set by the condition � &2E�ðtnow=tLSÞ2=3. Furthermore, we must also require that � not exceed the cutoff scale fG, or be smaller than the lightestmode in the tower. Once again, we find that the resulting integral expressions can be written in terms of the functionsIiðm; n; �;�; �min; �maxÞ defined in Eq. (4.38):

dn�dE�

��now� 12G�

2


ptnow

Mc

�snowsLS

��t1=2MRE

t2LS

�X4i¼1

Iiðmi � 5=2; ni; �i; �i; �XRBi�1 ; �

XRBi Þ; (4.50)

where the �XRBi are analogous to the �CMB

i appearing in Eq. (4.38). Determining the net contribution to the differentialphoton flux from decays of the a� for any particular choice of model parameters is thus simply a matter of numericallyevaluating Eq. (4.50).

We now turn to consider the observational limits on dn�=dE�. The diffuse extragalactic X-ray and gamma-ray

background spectra have been probed by a number of experiments. In the keV–MeV region, the most current data arethose from HEAO, COMPTEL, XMM, and Chandra; at energies above this, the most current data are those from EGRETand FERMI. Over this entire energy range, the diffuse photon spectrum is well modeled by a set of power-law fits, and thenonobservation of any discernible, sharp features in this spectrum imposes constraints on late relic-particle decays tophotons. For the data from the COMPTEL instrument, the best power-law fit is found to be [54]

dn�dE�

¼ 10:5 10�4

�E�

5 MeV

��2:4MeV�1 cm�1 s�1 str�1 800 keV & E� & 30 MeV; (4.51)

while the best fit to the HEAO data is found to be [53]

dn�dE�

¼

8>>>>>>>>>><>>>>>>>>>>:

7:88 103E�

keV

�1:29e�ðE�=41:13 keVÞ MeV�1 cm�1 s�1 str�1 0:1 keV & E� & 60 keV

0:43

�E�

60 keV

��6:5 þ 8:4

�E�

60 keV

��2:58 þ 0:38

�E�

60 keV

��2:05MeV�1 cm�1 s�1 str�1 60 keV & E� & 160 keV

3:8 105 E�

keV

�2:6MeV�1 cm�1 s�1 str�1 160 keV & E� & 350 keV

2:0 103E�

keV

�1:7MeV�1 cm�1 s�1 str�1 350 keV & E� & 2 MeV:

(4.52)

The Chandra satellite has improved upon these diffuse X-ray background constraints in the 1 keV & E� & 8 keV range byresolving a large fraction (� 80%) of this background into point sources. The residual spectrum in this region is wellrepresented by the power law [57]


055013-28

dn�dE�

¼ 2:6 103�E�

keV

��1:5MeV�1 cm�1 s�1 str�1 1 keV & E� & 8 keV: (4.53)

In the gamma-ray region, the most stringent current limits are those from EGRET and FERMI. Data on the diffuseextragalactic gamma-ray background from the former instrument [58] are reliable for photon energies within the range30 MeV & E� & 1:41 GeV, for which we find the best fit

dn�dE�

¼ 7:35 10�3

�E�

MeV

��2:35MeV�1 cm�1 s�1 str�1 30 MeV & E� & 1:41 GeV: (4.54)

Note that data exist for higher photon energies as well, but given that EGRET’s energy resolution is not as good at suchhigh energies, and given that these data have been superseded by data from FERMI, we do not use them in computing thispower-law fit. As for the FERMI data, they are well modeled by the power law [59]

dn�dE�

¼ 9:59 10�3

�E�

MeV

��2:41MeV�1 cm�1 s�1 str�1 274 MeV & E� & 70:7 GeV: (4.55)

In Fig. 7, we show a set of curves (solid colored lines)depicting the total contribution to the diffuse gamma-raybackground from the decaying a� fields, as given inEq. (4.50), for several different values of fX within therange 106–1016 GeV. Results are shown for �G ¼ 1 GeV(upper left panel), �G ¼ 1 TeV (upper right panel), and�G ¼ 100 TeV (lower panel). In each case, we have takenMc ¼ 10�11 GeV, TRH ¼ 5 MeV, and � ¼ gG ¼ ¼ 1.In addition, we have chosen a value for HI sufficientlylarge that none of the curves shown is significantly affectedby the ‘‘inflating away’’ of heavy modes which beginoscillating before inflation ends. In addition to thesecurves, we also display contours corresponding to theupper limits on the diffuse X-ray and gamma-ray fluxes(black dashed lines) given in Eqs. (4.51), (4.52), (4.53),(4.54), and (4.55). Any choice of model parameters for

which the differential photon flux dn�=dE� exceeds anyone of these observational-limit contours for any value ofE� is excluded. The results shown in Fig. 7 indicate thatwhile it is not trivial to satisfy these observational limits inour bulk-axion DDM model, the contributions to the dif-fuse X-ray and gamma-ray fluxes from a� decay are indeedsufficiently small that these limits are satisfied when �G islarge.We now consider the contribution to dn�=dE� from the

population of axions generated by their interactions withSM fields in the thermal bath after inflation. The contribu-tion to the diffuse-photon-flux spectrum dn�=dE� gener-

ated by such a population of axions is once again given byEq. (4.48), but with ��ðtLSÞ in Eq. (3.8) now replaced by

��ðtLSÞ � �T3LStLS

Z TRH

TMRE

3

ðTÞ�TLS

T

�3=ðTÞ gsðTLSÞ

gsðTÞ ½CID� ðTÞ þ CPrim

� ðTÞe�ð�þmeÞ=T�dT; (4.56)

as follows from Eq. (3.42). To derive an estimate for theexpected contribution to dn�=dE� from the resulting equa-tion, we proceed in essentially the same way as we did incalculating the contribution from axions produced viavacuum misalignment. The results of this calculation areshown in Fig. 8 for parameter values within or near thepreferred region of parameter space for our bulk-axionDDM model. Specifically, we have taken Mc ¼10�11 GeV, �G ¼ 1 TeV, TRH ¼ 5 MeV, and � ¼ gG ¼1. The solid colored curves shown correspond to severaldifferent choices of fX ranging from fX ¼ 1012 GeV tofX ¼ 1015 GeV. Once again, the dashed black lines indi-cate the observational limits on additional contributions todn�=dE�. It is clear from Fig. 8 that while the contributionto the diffuse X-ray flux from thermal axions within ourpreferred region of parameter space is certainly not negli-gible, it is also consistent with current observational limits.

We therefore conclude that even after the contribution fromthermal axions is included, our bulk-axion DDM model isconsistent with X-ray and gamma-ray data.

H. Axion decays and big-bang nucleosynthesis

The accord between the primordial abundances of lightnuclei inferred from observation and the predictions forthose abundances within the framework of standard BBNhas been one of the greatest triumphs of theoretical cos-mology. However, these predictions depend sensitively onthe cosmological parameters during the nucleosynthesisepoch. For example, the presence of additional relativisticdegrees of freedom in the thermal bath during BBN cansubstantially distort the abundances of the light elementsaway from their observed values. In addition, the decays ofunstable particles during or after the BBN epoch can alsoalter these abundances via the injection of both entropy and


055013-29

energy into the thermal bath. We must therefore ensure thatthe collective effects of a� decays in our model are suffi-ciently small so as not to disrupt the successful generationof light-element abundances via standard BBN.

Limits on the abundance of a single unstable relic par-ticle � from BBN are typically phrased as bounds on thenumber density n� that � would have at present time if it

were absolutely stable. In general, the BBN bound on n�for any given relic particle depends on the lifetime �� of

that particle. The most stringent limits are obtained forlifetimes �� Oð109–1010 sÞ, for which the correspond-

ing constraint is roughly [60,61]

m�

n�n�

& 10�13 GeV; (4.57)

where n� � 410:5 cm�3 denotes the present-day number

density of photons. This limit can also be written in theform

� � & 1:7 10�5; (4.58)

where �� is the relic abundance that � would have at

present time if it were absolutely stable.

G 1 GeVMc 10 11 GeV

f X 106 GeV

f X 108 GeV

f X 1010 GeV

f X 1012 GeV

f X 1014 GeV

f X 1016 GeV

10 5 0.001 0.1 10 1000 10510 35

10 25

10 15

10 5

105

1015

E MeV

dndE

MeV

1 cm2 s

1 str

1

G 1 TeVMc 10 11 GeV

f X 106 GeV

f X 108 GeV

f X 1010 GeV

f X 1012 GeV

f X 1014 GeV

f X 1016 GeV

10 5 0.001 0.1 10 1000 10510 35

10 25

10 15

10 5

105

1015

E MeV

dndE

MeV

1 cm2 s

1 str

1

G 100 TeVMc 10 11 GeV

f X 106 GeV

f X 108 GeV

f X 1010 GeV

f X 1012 GeV

f X 1014 GeV

f X 1016 GeV

10 5 0.001 0.1 10 1000 10510 35

10 25

10 15

10 5

105

1015

E MeV

dndE

MeV

1 cm2 s

1 str

1

FIG. 7 (color online). The diffuse-photon-flux spectrum dn�=dE� produced from axion decays in our bulk-axion DDM model with�G ¼ 1 GeV (upper left panel), �G ¼ 1 TeV (upper right panel), and �G ¼ 100 TeV (lower panel). Each solid colored curvecorresponds to a different choice of fX within the range 106–1016 GeV. In all panels, we have taken Mc ¼ 10�11 GeV, HI ¼ 1 GeV,TRH ¼ 5 MeV, and � ¼ gG ¼ ¼ 1. By contrast, the dashed black contours represent the upper bounds on dn�=dE� derived from

observational limits on the diffuse photon flux using a number of instruments sensitive in the X-ray and gamma-ray regions. As evidentfrom these plots, the diffuse-photon-background contribution arising from axion decay in our bulk-axion DDM model is consistentwith all observational limits when �G is large.


055013-30

Once again, however, as with other constraints on tradi-tional models of decaying dark matter (such as those fromthe CMB and the diffuse X-ray and gamma-ray back-grounds), these constraints are not readily applicable tomodels within the DDM paradigm, since the dark-mattercandidate in these models is an ensemble with no single,well-defined mass or lifetime. Thus, we must reexaminethe derivation of the BBN constraints on decaying relicparticles in order to determine what restrictions these con-siderations place on the parameter space of our bulk-axionDDM model. While a detailed calculation of the preciselimits BBN considerations impose on DDM scenarios, ingeneral, is beyond the scope of this paper, it is straightfor-ward to demonstrate that BBN constraints do not signifi-cantly restrict the parameter space of the particular modelwhich concerns us here.

We begin by noting that in traditional, single-particledark-matter scenarios, an unstable dark-matter candidate �with a relic abundance�� CDM is generally consistent

with all astrophysical and cosmological limits on dark-matter decays, provided that �� * �min

� � 1026 s [56]. It

therefore follows that any a� in the DDM ensemble with alifetime �� * �min

� will have no impact on BBN within

regions of parameter space in which the WMAP constraint�tot � �CDM on the total dark-matter relic abundance issatisfied. Thus, we may safely conclude that our bulk-axionmodel of dynamical dark matter is consistent with BBNconstraints within such regions of parameter space, pro-vided that

� tot & 1:7 10�5; (4.59)

where �tot denotes the collective contribution which the

set of a� with lifetimes �� < �min� would have made to the

dark-matter relic abundance at present time if they wereabsolutely stable. In other words, the BBN constraint weare imposing in Eq. (4.59) effectively rests upon the ex-tremely conservative approach of treating all states in theDDM ensemble whose lifetimes are less than �min

� as if they

had lifetimes �� which are in the range which is mostdangerous for BBN, namely �� 109–1010 s. We empha-size that while this criterion is a sufficient condition forsuccessful BBN, it does not represent the true BBN con-straint, which is always far less stringent.

In Fig. 9, we display contours of �tot for a DDM

ensemble of photonic axions with c� ¼ 1, as a function

of fX, Mc, and �G. The left panel shows the results for�G ¼ 1 GeV, the center panel for �G ¼ 1 TeV, and theright panel for �G ¼ 100 TeV. In each case, we havetaken TRH ¼ 5 MeV, HI ¼ 100 TeV, and � ¼ gG ¼ ¼1. In each panel of Fig. 9, we see that the criterion inEq. (4.59) is amply satisfied throughout essentially theentire region of parameter space shown. It therefore fol-lows that our bulk-axion model is consistent with success-ful BBN throughout this region of parameter space.

I. Axion decays and late entropy production

One additional physical consequence of the late decaysof unstable relic particles is the generation of entropy asthose particles ‘‘dump’’ their energy density into the radia-tion bath. Indeed, a number of considerations placeconstraints on late entropy production from decaying par-ticles. For example, late entropy generation can upset the

f X 1012 GeV

f X 1013 GeV

f X 1014 GeV

f X 1015 GeV

Mc 10 11 GeVG 1 TeV

10 5 0.001 0.1 10 1000 10510 35

10 25

10 15

10 5

105

1015

E MeV

dndE

MeV

1 cm2 s

1 str

1

f X 1012 GeV

f X 1013 GeV

f X 1014 GeV

f X 1015 GeV

Mc 10 11 GeVG 100 TeV

10 5 0.001 0.1 10 1000 10510 35

10 25

10 15

10 5

105

1015

E MeV

dndE

MeV

1 cm2 s

1 str

1

FIG. 8 (color online). The diffuse-photon-flux spectrum dn�=dE� produced from the decays of a population of a� produced byinteractions among the SM particles in the thermal bath after inflation. The left panel shows the results for �G ¼ 1 TeV, while theright panel shows the results for�G ¼ 100 TeV. In each case, we have takenMc ¼ 10�11 GeV, TRH ¼ 5 MeV, and � ¼ gG ¼ 1. Thesolid colored curves indicate the diffuse-photon-flux contributions corresponding to different choices of fX. As in Fig. 7, the dashedblack contours indicate the upper bounds on dn�=dE� derived from observational limits on the diffuse X-ray and gamma-ray fluxes,

and we see that our model is consistent with these bounds.


055013-31

light-element predictions from standard BBN and produceobservable features in the CMB. In this section, we exam-ine the effect of the late decays of the a� on the entropydensity of the Universe in our bulk-axion DDM model as afunction of time in order to verify that no perceptibleeffects can arise which might serve to exclude our model.

During any given epoch, the entropy density of theUniverse is dominated by the contribution from radiationand therefore well approximated by

s � Xi

4�iðTiÞ3Ti

¼ �2

30gsðTÞT3; (4.60)

where the index i runs over all relativistic particle species,Ti is the temperature associated with any particular suchspecies, and gs is the number of interacting degrees offreedom at temperature T. During the early stages of thehistory of the Universe (prior to neutrino decoupling), allsuch species are characterized by a common temperatureTi � T. During such epochs, gsðTÞ � gðTÞ, and the en-tropy density is therefore directly proportional to the totalenergy density �rad of radiation. Indeed, even during sub-sequent epochs, gs and g remain roughly similar, and �rad

remains a good indicator of the entropy density. Thus, byevaluating the contribution to �rad from a� decays in ourbulk-axion DDM model, we can assess the effect of thesedecays on both the energy and entropy densities of theUniverse.

In the LTR cosmology, as in the standard cosmology,�rad evolves according to an equation similar to Eq. (4.30):

d�rad

dt¼ �4H�rad þ �� þX

�

BRðradÞ� ��: (4.61)

This equation assumes the presence of a tower of decaying

a�, where BRðradÞ� is the total branching fraction of a� into

relativistic particles. Note, however, that since we areworking within the context of LTR cosmology, the effectsof inflaton decays on the energy and entropy densities ofthe Universe remain relevant until very late times t� tRH.Thus we have explicitly included an additional source term�� in Eq. (4.61) to account for the effect of such inflaton

decays, where �� and �� respectively denote the decay

rate and energy density of the inflaton field �.The contribution to �rad from inflaton decays can readily

be calculated from standard results pertaining to the LTRcosmology (for a review, see, e.g., Ref. [16]). As theUniverse exits the inflationary epoch at a time tI �2=ð3HIÞ, the energy density stored in the inflaton field isinitially �� ¼ �crit ¼ 3H2

IM2P. During subsequent epochs,

the inflaton source term for radiation is approximatelygiven by

�� 3H2IM

2P

2tRH

�tIt

�e�t=2tRH ; (4.62)

where is defined as in Eq. (3.3), and we have used the factthat the inflaton-decay rate is related to the reheating timeby �� 1=ð2tRHÞ. Note that this source term is negligible

at times t � tRH, when by definition �� rad, and hence

can safely be neglected at such times. By contrast, at earlytimes t & tRH, the inflaton source term is expected todominate in Eq. (4.61), in the sense that

�� X�

BRðradÞ� ��: (4.63)

FIG. 9 (color online). Contours of the collective contribution to �tot from the set of a� with lifetimes �� < �min

� for a DDM ensembleof photonic axions with c� ¼ 1. The left, center, and right panels display the results for �G ¼ 1 GeV, �G ¼ 1 TeV, and �G ¼100 TeV, respectively. In each case, we have taken TRH ¼ 5 MeV, HI ¼ 100 TeV, and � ¼ gG ¼ ¼ 1. In each panel, we see thatBBN constraints are amply satisfied throughout essentially the entire region of parameter space shown.


055013-32

Whenever this condition is satisfied, the contribution to�rad from a� decays is inconsequential compared to thatfrom inflaton decays, and the axion source term can there-fore safely be neglected.

In assessing the contribution from a� decay, we onceagain choose to focus on the case of a photonic axion withc� ¼ 1; this implies that the decay mode a� ! �� domi-

nates the contribution to �rad. In this case, the source termfor radiation due to a� decay is just the source term forphotons given in Eq. (4.37). In this case, solving Eq. (4.61)for �rad, we find that

�radðtÞ ¼ ��radðtÞ þZ t

tG

�t0

t

�4=3X

�

BRð2�Þ� ��ðt0Þdt0;

(4.64)

where ��radðtÞ is the solution for �radðtÞ in the absence ofany additional contribution from a� decays. Once againmaking use of the integral functions Iiðm; n; �; �;�min; �maxÞ defined in Eq. (4.35) to approximate the sumover modes, we obtain

�radðtÞ ¼ ��radðtÞ þ2G�

2

Mc

Z t

tG

dt0t0=3

t4=3

X4i¼1


i Þ 8><>:t1=2RH t & tRH1 tRH & t & tMRE

t1=2MRE t * tMRE:

(4.65)

In Fig. 10, we show how the contribution to �rad from a�decays in our bulk-axion DDM model evolves with timefor a variety of different choices of model parameters. Theleft panel shows results for �G ¼ 1 GeV, the center panelfor �G ¼ 1 TeV, and the right panel for �G ¼ 100 TeV.The solid colored curves in each panel correspond todifferent choices of fX within the range 1010–1016 GeV.For all curves shown, we have assumed a photonic axionwith c� ¼ 1, and we have takenMc ¼ 10�11 GeV, TRH ¼5 MeV, HI ¼ 100 TeV, and � ¼ gG ¼ ¼ 1. The blackdashed curve represents the total value of �rad, whichincludes the standard contribution from inflaton decaysduring the reheating epoch. Since such inflaton decaysconstitute the dominant source for radiation prior to theend of reheating, the range of times shown in each panel

extends from tRH to present time. The value ofHI has beenchosen here to be sufficiently large that the effect ofheavier a� with � * 3HI=2 being inflated away is unim-portant. Note, however, that for significantly smaller valuesof HI, the contribution to �rad from axion decays can befurther suppressed by this effect.The differences among the curves shown in Fig. 10 for

different choices of fX and �G ultimately stem from theeffects of axion mixing on the abundances �� and decaywidths �� of the individual axion modes. The resultsshown in the left panel correspond to the case in which

�G is sufficiently small that y � 1 for all choices of fXshown. In this small-mixing regime, � * �trans for allbut the lowest-lying mode in the axion KK tower, and

Eqs. (2.12) and (3.8) imply that �� / f�2X and �� / f�2

X .

G 1 GeVMc 10 11 GeV

HI 100 TeV

f X 1010 GeV

f X 1012 GeV

f X 1014 GeV

f X 1016 GeV

100 106 1010 101410 70

10 55

10 40

10 25

10 10

105

t s

rad

GeV

4

G 1 TeVMc 10 11 GeV

HI 100 TeV

f X 1010 GeV

f X 1012 GeV

f X 1014 GeV

f X 1016 GeV

100 106 1010 101410 70

10 55

10 40

10 25

10 10

105

t s

rad

GeV

4

G 100 TeVMc 10 11 GeV

HI 100 TeV

f X 1010 GeV

f X 1012 GeV

f X 1014 GeV

f X 1016 GeV

100 106 1010 101410 70

10 55

10 40

10 25

10 10

105

t s

rad

GeV

4

FIG. 10 (color online). The total contribution to the radiation-energy density �rad from photonic a� decays in our bulk-axion DDMmodel (solid lines), plotted as functions of time for a variety of different choices of fX. The left panel shows the results for �G ¼1 GeV, the center panel for �G ¼ 1 TeV, and the right panel for �G ¼ 100 TeV. In each case, we have assumed a photonic axionwith � ¼ gG ¼ ¼ 1, and we have takenMc ¼ 10�11 GeV, TRH ¼ 5 MeV, andHI ¼ 100 TeV. Also shown in each panel is the totalvalue of �rad as a function of time in the LTR cosmology (black dashed line), which includes the contribution from inflaton decay. In allcases, the collective contribution to �rad from a� decays at all times t < tnow remains negligible compared to the primordialcontribution generated via inflaton decays during reheating. Thus our bulk-axion DDMmodel does not lead to overproduction of eitherradiation-energy density or entropy during any prior cosmological epoch.


055013-33

It therefore follows that the photon source term

BRð2�Þ� �� associated with each a� within this regime

decreases uniformly and substantially with increasing fX,as indicated. By contrast, as �G is increased, several com-peting effects play an increasingly important role in deter-

mining the magnitude of BRð2�Þ� �� for certain �. This is

because �trans increases with increasing �G; hence forlarge �G a greater number of the a� are brought into the

� & �trans regime, in which �� / f2X and �� / f2X.

Increasing fX therefore has the effect of increasing theinitial magnitude of the photon source terms associatedwith the a� in this regime. However, the lifetimes of these

modes also increase with increasing fX, and hence thetransfer of their energy density to radiation is deferred untillater times, when �rad is smaller and the contribution froma� decays can have a proportionally greater impact. Theinterplay between these effects results in the behaviorshown in the right two panels of Fig. 10.

Note that the curves for the total energy density shown inFig. 10, which are dominated by the contribution frominflaton dynamics, drop more rapidly as a function oftime than the contributions from axion dynamics. Thisreflects the continuing generation of new radiation-energydensity from the ongoing decays of the individual a�within our DDM ensemble. In all cases, however, thecollective contribution to �rad from a� decays at all timest < tnow remains negligible compared to the primordialcontribution generated via inflaton decays during reheat-ing. Thus our bulk-axion DDM model does not lead tooverproduction of either radiation-energy density or en-tropy during any prior cosmological epoch.

J. Vacuum energy and overclosure

In traditional dark-matter scenarios involving a single,stable dark-matter candidate �, the dark-matter relic abun-dance �� increases monotonically up to and beyond the

present time. As a result, verifying that �� satisfies

WMAP constraints at the present time is sufficient toguarantee that � does not overclose (or prematurelymatter-dominate) the Universe at all previous times aswell. However, one of the hallmarks of the DDM scenariois that this is no longer true: although �tot likewise expe-riences a Hubble-driven growth during the earliest phasesof the evolution of the Universe, this quantity can never-theless drop during later epochs. This is possible within theDDM framework because the single, stable dark-mattercandidate � characteristic of most traditional dark-matterscenarios is replaced by a complex, multicomponent dark-matter ensemble whose constituents can have a broadspectrum of lifetimes and abundances. As a result, thedecays of certain dark-matter components within the en-semble can cause�tot to decline—even prior to the presentday. Indeed, such behavior for �tot can be quite dramatic,and is illustrated in Fig. 6 of Ref. [1] for the special case in

which the DDM ensemble consists of a KK tower ofdecaying dark fields. Thus, within the DDM framework,it is no longer sufficient to verify that �tot satisfies over-closure constraints at the present time; we must also verifythat it has satisfied such overclosure constraints (and con-straints from premature matter- or vacuum-energy domi-nation) at all prior moments during the history of theUniverse.It turns out, however, that this is not a problem in our

bulk-axion DDM model. Since our model already satisfiesWMAP constraints at present time within our preferredregion of parameter space [2], it can run afoul of over-closure constraints in the past only if the negative rate ofchange of �tot is sufficiently great that �tot might haveexceeded unity within the past history of the Universe.However, as discussed in Refs. [1,2], this rate of changeis described by an effective equation-of-state parameterweff , and two things are already known about the value ofthis parameter in our model: first, it is extremely small atthe present day, i.e., 10�23 & weff & 10�12 [2], and sec-ond, it was even smaller in the past. Indeed, this latterassertion follows from the generic behavior of weff shownin Fig. 8 of Ref. [1]: for a generic KK tower,weff reaches itsmaximum at the present day and is exponentially smallerprior to this time. Thus, working backwards from thepresent epoch, and given the finite age of the Universe,we see that it is not possible for �tot to have violatedoverclosure bounds at any point during the history of theUniverse.One related concern which arises in our bulk-axion

DDM model, due to our reliance on the misalignmentmechanism for the generation of the primordial relic abun-dances of the a�, is the risk of premature vacuum domina-tion. Indeed, any a� for which t� > tG will contribute to thetotal dark-energy abundance �vac during the period whentG & t & t�, within which its energy density �� is non-vanishing but before which it begins oscillating. Since ��

remains constant during this period, the contribution to�vac scales like �� / t2 during any MD or RD epoch.Since this represents a rate of increase far faster than thatassociated with matter or radiation, the threat of prematurevacuum domination from fields which remain as vacuumenergy for a long duration is of particular concern. Indeed,in extreme cases, such fields could potentially give rise toan additional period of inflation, leading to gross incon-sistencies with the predictions of BBN, CMB data, and soforth.In our bulk-axion model, however, it is straightforward

to demonstrate that no such inconsistencies with observa-tional data arise. The masses of all of the a�, with the soleexception of the zero mode a0, are bounded from below bythe Newton’s-law-modification constraint in Eq. (4.1),since �i � Mc=2 for i > 0. For all such modes with t� >tG, this constraint on � implies a bound t� > 6:7510�14 s on the oscillation-onset time of the mode. (The


055013-34

remaining modes, for which tG ¼ t�, never contribute to�vac.) This time scale is sufficiently early that the collec-tive vacuum-energy contribution from these a� poses nothreat of overclosure or premature vacuum domination.The �� contributions from these fields simply do nothave time to grow to a problematic size.

This leaves only the contribution from a0, whose oscil-lation time scale can be substantially longer than the upperlimit quoted above for the higher modes in situations inwhich y � 1. Since A�0

� 1 in this limit, Eq. (3.8) implies

that prior to the time t�0at which it begins oscillating, the

relic abundance of a0 is given by

��0� 3

2

m2Xf

2X

M2P

�t

�2: (4.66)

Therefore, one finds that by the time of oscillation, whichis given by t�0

� �0=2mX in this limit, ��0

will have

grown to

��0ðt�0

Þ � 3

8

f2XM2

P

: (4.67)

This result is independent of mX, and implies that thecontribution of the a0 to �vac is not a cause for concern

for sub-Planckian values of fX. Indeed, this is to be ex-pected: in this regime, a0 functions effectively like a four-dimensional axion. Early vacuum-energy domination isknown not to be a problem for light axions and axionlikeparticles (see Ref. [37] and references therein) in purelyfour-dimensional theories.

K. Misalignment production and isocurvatureperturbations

In an inflationary cosmology, fluctuations in the energydensity of any population of particles produced ther-mally—i.e., via rapid interactions in the radiation bathduring the reheating phase—stem from the primordialperturbations in the energy density of the inflaton field.Consequently, such fluctuations represent spatial variationsin the total energy density, but not in the relative contribu-tions of individual particle species to that total density.Such variations, in turn, imply fluctuations in the localspacetime curvature and are therefore sometimes also re-ferred to as curvature perturbations. By contrast, fluctua-tions in the energy density of any population of particlesproduced via means uncorrelated with the inflaton field(and therefore nonthermal) can also give rise to fluctua-tions of the isocurvature type—i.e., perturbations in therelative contributions of different species to the total en-ergy density, with that total energy density held fixed.Recent WMAP observations of the CMB power spectrum,taken in combination with baryon acoustic oscilla-tion (BAO) measurements and supernova data, place astringent bound [10] on any deviations from adiabaticityin primordial energy-density fluctuations. This bound is

typically expressed in terms of the fractional contribution�0 to the CMB power spectrum from axion isocurvatureperturbations:

�0 � hð�T=TÞ2isoihð�T=TÞ2toti

< 0:072; (4.68)

where hð�T=TÞ2toti and hð�T=TÞ2isoi respectively denote the

total average root-mean-squared fluctuation in the CMBtemperature, and the average root-mean-squared tempera-ture fluctuation due to isocurvature perturbations alone.Since the a� fields which compose our dynamical dark-matter ensemble are presumed to be produced nonther-mally, via the misalignment mechanism, it is necessaryto investigate the implications of this bound for our model.Our discussion of isocurvature perturbations in our bulk-

axion DDM model in large part parallels the discussion ofsuch perturbations in traditional QCD-axion models pre-sented in Ref. [62], to which we refer the reader for a morecomplete introduction and discussion of the formalism andmethodologies used. It turns out to be convenient to ex-press the fluctuations of any given a� in terms of thefractional change S� in the ratio of its number density n�to the entropy density s of the Universe. This quantity canbe written in the form

S� � �ðn�=sÞðn�=sÞ ¼ �n�

n�� 3

�T

T: (4.69)

We assume that the production of all other particle speciesc i (i.e., the SM fields) ultimately results from inflatondecay, and that the density fluctuations for these speciesare purely adiabatic, with Si ¼ 0. Since, by definition, thefluctuation �� in the total energy density vanishes forisocurvature fluctuations, it therefore follows that thesum of the fluctuations in the energy densities of thevarious particle species obeys a constraint which may bewritten in the form

X�

��

�S� þ 3

�T

T

�þ 3

Xi

�i

�T

Tþ 4�rad

�T

T¼ 0; (4.70)

where the �i denote the energy densities associated withmassive species other than the a�, and �rad once againdenotes the total energy density of radiation. In our bulk-axion DDM model, the abundances of all of the a� aredetermined by a single misalignment angle . As discussedin Ref. [1], this reflects the ultimate five-dimensional na-ture of the axion field. This in turn implies that the densityfluctuations �n� for all of these fields are determined bythe fluctuations � in this misalignment angle generated byquantum fluctuations during inflation. The fact that thefluctuations �n� are all determined by � implies thatthe S� � S are essentially equal for all a�; henceEq. (4.70) simplifies to

�totS ¼ �3

��mat þ 4

3�rad

��T

T; (4.71)


055013-35

where �mat denotes the total abundance of matter in theUniverse, including the contributions from baryonic mat-ter, the ensemble of dark axions, and any other particleswhich might contribute to the dark-matter relic abundance,and �rad is the relic-abundance contribution from radia-tion. This expression is identical to that which describesthe isocurvature perturbations associated with a single,four-dimensional axion. Therefore, assuming that the fluc-tuations in are Gaussian, it follows that in our axionDDM model, �0 is given by the standard expression [62]

�0 ¼ 8

25

��

tot

�mat

�2 1

hð�T=TÞ2toti 2

ð22 þ 2Þ

ð2 þ 2Þ2

; (4.72)

where �mat denotes the present-day value of �mat, and

where 2 � hð�Þ2i denotes the variance associated with

fluctuations in .This result makes intuitive sense. Although our DDM

model has essentially partitioned the total dark-matterabundance amongst a large number of different KK axionfields, the underlying five-dimensional nature of the KKtower has correlated the individual fluctuations of thesefields so that they are governed by the fluctuation of asingle misalignment angle . It is therefore not a surprisethat the expected magnitude for isocurvature fluctuationsin our model turns out to be no greater than it is in standard,four-dimensional axion models.

All that remains, then, for us to do in order to determinethe value of �0 in our bulk-axion DDM model, is to assessthe magnitude of 2

. Assuming again that the fluctuations

in are Gaussian, this quantity is given by

2 ¼

H2I

4�2f2X: (4.73)

Since we are operating within the context of an LTRcosmology with TRH �OðMeVÞ, as discussed above, it isby no means problematic (and in fact quite natural) for

HI � fX. Therefore, as long as �Oð1Þ, as might beexpected from naturalness considerations, it can safely beassumed that � . Substituting into Eq. (4.72) theexperimentally observed [10] values hð�T=TÞ2toti � ð1:110�5Þ2 and �

mat � 0:262 we find that �0 is well approxi-mated by

�0 � 1:95 109�HI�

tot

fX

�2

(4.74)

in our bulk-axion model. Combining this result with theupper bound on �0 quoted in Eq. (4.68) yields the con-straint

HI & 6:07 10�6

�fX�

tot

�: (4.75)

We consider the case in which �tot � �CDM and in which

the axion ensemble is responsible for essentially the en-tirety of the observed dark-matter relic abundance. This

corresponds to fX � 1014–1015 GeV. We then find that for�Oð1Þ, the resulting constraint HI & 109–1010 GeV onthe Hubble parameter during inflation is relatively mild.Indeed, there is no difficulty in satisfying this constraint ineither the standard or the LTR cosmology. We thus con-clude that isocurvature perturbations do not present anyproblem for our bulk-axion model of dynamical dark mat-ter. Moreover, a low scale forHI can be regarded as naturalin the context of an LTR cosmology.It is worth remarking, however, that the above results

have implications for the detection of primordial gravita-tional waves. Limits on primordial gravitational wavesfrom observations of the CMB can be conveniently pa-rametrized in terms of the scalar-to-tensor ratio r. Forexample, consider single-field models of inflation, inwhich r ¼ 16�, where � ¼ M2

PðV 0=VÞ2=ð4�Þ is the infla-ton slow-roll parameter, with V and V0 denoting the in-flaton potential and its first derivative with respect to theinflaton field, respectively [63]. In the context of our bulk-axion DDM model, the standard relation (see, e.g.,Ref. [10]) between r and �0 takes the form

r ¼ 22f2XM2

P

��CDM

�tot

�2 �0

1� �0

: (4.76)

As discussed above, consistency with the bounds inEqs. (4.68) and (4.75) requires that HI � 2�fX and�0 � 1. In this regime, one finds that the expectedtensor-to-scalar ratio is essentially independent of �

tot

and well approximated by

r � 2:7 108�HI

MP

�2: (4.77)

Current WMAP observations, again in conjunction withBAO and supernova data, place an upper bound r < 0:22on the tensor-to-scalar ratio [10]. Thus, Eq. (4.77) results ina constraint HI & 6:7 1013 GeV on the Hubble scaleduring inflation—a constraint which Eq. (4.75) implies isalready automatically satisfied, even forOð1Þ values of themisalignment angle . The upshot is therefore that whilethere is no conflict between current limits on isocurvatureperturbations and the predictions of our bulk-axion DDMmodel, the requirement that HI be relatively small in thismodel suggests that r should likewise be quite small—atleast in the simplest of inflationary scenarios. Constraintson the spectral index ns from WMAP [10] can simulta-neously be satisfied for small r without difficulty, forexample in negative-curvature models of inflation, whichtend to predict small r [64].In summary, we conclude that current constraints on

isocurvature perturbations can be satisfied in our bulk-axion DDM model without too much difficulty. However,we note that any conclusive measurement of r within thesensitivity range of the Planck satellite would have severeramifications for this model.


055013-36

L. Axion abundances and quantum fluctuationsduring inflation

Thus far in this paper, we have disregarded the effects ofthe quantum fluctuations that naturally arise for any mass-less or nearly massless field during the inflationary epoch.In particular, the low-momentum modes of any a� in ourmodel with a mass � & HI have wavelengths which ex-ceed the Hubble length during inflation; excitations of suchlow-momentum modes are therefore indistinguishablefrom a VEV and consequently do not inflate away. Theseexcitations necessarily yield a primordial energy-densitycontribution in our bulk-axion DDM model which cannotbe avoided in any inflationary cosmology. Consistencywith the relic-abundance predictions discussed inSect. III A therefore requires that this primordial energydensity be small compared to that which results frommisalignment production.

In particular, it is possible to formulate a condition thatensures that these quantum fluctuations not invalidate ourprevious analysis. Clearly, one criterion that any suchcondition must enforce is that such fluctuations not havea significant effect on the total relic abundance of theensemble. We may formulate this constraint as a require-ment that the difference between the full present-day relic

abundance ~�tot, which incorporates the effect of these

fluctuations, and the result �tot obtained in the absence

of such corrections be negligible—i.e., that

j ~�tot ��

totj � �tot: (4.78)

While the condition in Eq. (4.78) is certainly a necessaryone, it is not by itself sufficient to ensure that vacuumfluctuations during inflation do not lead to phenomenologi-cal difficulties for our model. This is because within theDDM framework, dark-matter stability is not a require-ment, and consistency with observational constraints isarranged by balancing decay widths against abundancesacross the entire dark-matter ensemble. Indeed, as we havedemonstrated, misalignment production provides preciselythe right relationship between the �� and �� to mitigatethe deleterious effects of the heavier, more unstable statesin our ensemble and render our model phenomenologicallyviable. We must therefore ensure that this delicate balanceis not disrupted by the effects of vacuum fluctuationsduring inflation.

Within the preferred region of parameter space of ourbulk-axion DDM model, as discussed in Sect. III A, theoscillation-onset times for the lighter a� in the tower arestaggered in time. As a result, these lighter modes collec-tively dominate in�tot. It therefore follows that whether ornot the total-relic-abundance constraint in Eq. (4.78) issatisfied depends primarily on how vacuum fluctuationsaffect the abundances of these most abundant modes alone.By contrast, the balancing of lifetimes against abundancesdepends on the properties of the full KK tower, and notmerely on the attributes of the lighter modes which domi-

nate �tot. The corresponding condition we impose on ourmodel therefore represents an even stronger constraint thanthe one appearing in Eq. (4.78) and indeed subsumes it. To

wit, we require that the full relic abundance ~�� of eachaxion mode not differ significantly from the correspondingabundance �� obtained in the absence of corrections dueto vacuum fluctuations during inflation—i.e., that

j ~�� j � �� for all �: (4.79)

We emphasize that this is an overly conservative con-straint, and that consistency with observational data iscertainly possible even if vacuum fluctuations do have asignificant effect on the abundances of certain a�.However, as we shall demonstrate, the restriction thatthis overly conservative constraint imposes on our model(which primarily turns out to take the form of an upper

bound on HI for any allowed choice of fX,Mc, and �G) isnot terribly severe.In order to determine how this condition restricts the

parameter space of our model, we must first assess whateffect vacuum fluctuations during inflation have on theindividual energy densities �� and relic abundances ��

of the constituent fields in our dark-matter ensemble. Webegin by noting a generic result in inflationary cosmologies(for a review, see Ref. [65]), namely that the variance h�2iin the amplitude of any light scalar � with a mass m� &

HI induced by vacuum fluctuations during inflation isgiven by

h�2i �H3I �tI4�2

; (4.80)

where �tI denotes the duration of inflation. A fluctuation ofthis order will therefore be induced in the amplitude of anyaxion in our dark-matter ensemble with a mass smallerthan HI. Moreover, we note that the relationship between�tI and HI is constrained by the fact that successful reso-lution of the smoothness and flatness problems requiresNe � HI�tI * 60, where Ne denotes the number ofe-foldings of inflation. In typical scenarios, Ne lies onlyslightly above this lower bound; hence �tI is typicallyexpected to be such that HI�tI �Oð100Þ. We will fre-quently express our results in terms of Ne in what follows.We begin our discussion of the effect of these fluctua-

tions on the abundances of the constituent particles in ourdark-matter ensemble by examining the simple case inwhich tG & tI. In this case, the axion mass-squared matrixattains its asymptotic, late-time form before inflation ends,and the a� are consequently already the axion mass eigen-states during the inflationary epoch. Thus, we find that thetotal energy density associated with each a� with � & HI

at the end of inflation is given by

��ðtIÞ � 1

2�2

�A�fX þ ��

HI

ffiffiffiffiffiffiNe

p2�

�2; (4.81)

where


055013-37

�� Oð1Þ � & HI

0 � * HI(4.82)

is a random coefficient which parametrizes the fluctuationin the field a�.

Before proceeding further, we remark that the aboveresults depend critically on the assumption that tG & tI.In other words, we have assumed that the instanton dy-namics associated with the gauge group G has alreadyoccurred and made its contributions to the KKmasses priorto the onset of the quantum fluctuations that arise due toinflation. By contrast, if tG * tI, the quantum fluctuationswill occur first, when the axion mass matrix is still diagonaland when the KK-momentum modes and mass eigenstatescoincide. In such cases, these are the modes which developquantum fluctuations, and the mode mixing induced by theinstanton dynamics occurs only later.

This distinction is important, because the resulting en-ergy density for each a� takes a somewhat different formwhen tG * tI:

�� ¼ 1

2�2

�X1n¼0

U�n

�fX�n;0 þ �n

HI


p2�

��2: (4.83)

In this expression, U�n is the mixing matrix in Eq. (2.14)and �n is the analogue of �� discussed above, with �n

taking nonzero values only when n & HI=Mc.A priori, this expression results in a different value for

�� than that in Eq. (4.81). However, it turns out that theeventual constraints associated with Eq. (4.83) are no morestringent than those which we shall eventually calculate forEq. (4.81). In order to understand why this is the case, let usconsider an even more dramatic situation in which �n

actually takes a fixed, positive value �� for all n—evenvalues of n beyond the inflationary cutoff HI=Mc. In thiscase, we can make use of the identity

X1n¼0

U�n ¼ fð~�ÞA�; (4.84)

where fð~�Þ � ð~�2 þ ffiffiffi2

p � 1Þ= ffiffiffi2

p, in order to rewrite

Eq. (4.83) in the form

�� ¼ 1

2�2

�A�fX þ fð~�Þ ��HI


p2�

�2: (4.85)

Remarkably, this is essentially the same expression as wewould have obtained from Eq. (4.81) when �� ¼ �� for all�, except that the fluctuation contribution now comes

multiplied by an extra ‘‘scaling’’ factor fð~�Þ. It is easy to

verify that fð~�Þ ! 1 as ~� ! 1, whereas for small ~� we

find that fð~�Þ � 1. This indicates that the effects of theinflation-related quantum fluctuations are actually sup-pressed for the lighter modes, relative to what occurs inthe case with tG & tI. The magnitude of this suppressiondepends on y, and is more severe when y � 1 (i.e., whenthe axion modes are more fully mixed). We thus conclude

that the contributions from the quantum fluctuations thatarise during inflation are greater when they occur after theinstanton dynamics turns on (and after the KK mode mix-ing), rather than before. We shall therefore concentrate onthe tG & tI case in what follows.Given the result in Eq. (4.81), we see that the effect of

vacuum fluctuations on the �� will be small for values of �which satisfy the condition

A�fX *HI


p2�

: (4.86)

Since A� is a monotonically decreasing function of �, itfollows that within any given tower of a�, there exists acritical mass value

�fluc � mXffiffiffi2

p264

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ �2

y2

�2 þ 32�22f2X

NeH2I

vuut ��1þ �2

y2

�3751=2

(4.87)

below which the effect of vacuum fluctuations on thecorresponding energy density �� is negligible. These ��

are therefore well approximated by Eq. (3.8), and the

corresponding abundances ~�� are given by Eq. (3.9) orEq. (3.10), depending on the value of t�. By contrast, for�fluc & � & HI, the effect of vacuum fluctuations over-whelms the effect of vacuum misalignment. The initialenergy density of each a� in this regime is thereforeeffectively set at tI and is approximately given by

��ðtIÞ � Ne

8�2�2H2

I : (4.88)

Since the Newton’s-law-modification bound on Mc inEq. (4.1) implies that t� & tRH for each such field, it there-fore follows that at all subsequent times, the correspondingrelic abundance is given by

~�� 3NeH2I

4�2M2P

e��ðt�tIÞ

8>>>>><>>>>>:

14 1=�& t& tRH

49

t

tRH

1=2

tRH & t& tMRE

14

tMRE

tRH

1=2

t* tMRE:

(4.89)

To summarize, we see that the axion KK tower separatesinto three distinct regimes within each of which different

physics plays a principal role in determining ~��. In the

� & �fluc regime, the effect of vacuum fluctuations on ~��

is negligible and the results in Sect. III A continue to hold.In the �fluc & � & HI regime, the opposite is true: vacuumfluctuations dominate and the abundances of the a� aregiven by Eq. (4.89). Finally, in the � * HI regime, thewavelengths of even the lowest-lying momentum modes ofeach a� fall short of the Hubble length during the infla-tionary epoch. Such modes therefore behave unambigu-ously like particles, and are consequently inflated away.


055013-38

We are now ready to address the constraint we have

imposed on the individual abundances ~�� in Eq. (4.79).Since the effect of vacuum fluctuations is negligible bothfor � * HI and for � & �fluc, it follows that this constraintwill be satisfied whenever HI � �fluc. Moreover, since�fluc itself decreases with increasing HI, as indicated inEq. (4.87), we find that our constraint may be expressed inthe formHI � Hcrit

I , whereHcritI is the value of the Hubble

parameter during inflation for whichHI ¼ �fluc. In Fig. 11,we display contours of Hcrit

I as a function of the model

parameters fX, Mc, and �G. For the large values of �G

characteristic of our preferred region of parameter space,we observe that the constraint in Eq. (4.79) is satisfied forHI � Hcrit

I �Oð10–100 GeVÞ. For smaller values of �G,although the constraint is certainly more severe, we never-theless observe that the bound can be satisfied for HI �Hcrit

I �Oð10–100 keVÞ. This condition on HI has non-trivial implications for the construction of explicit infla-tionary models, since values of HI of this magnitude tendto be rather nongeneric [66] among typical classes ofinflationary potentials. However, as discussed in Ref. [2],such a scale for HI is certainly not excluded (see, e.g.,Refs. [16,67]). Moreover, a small value forHI fits naturallywithin the context of the LTR cosmology.

M. Other astrophysical constraints on light axions

In addition to the constraints we have discussed above,there exist a number of additional astrophysical and cos-mological bounds on theories involving light axions andaxionlike particles. Indeed, particles of this sort can giverise to a number of potentially observable effects [9], suchas a rotation of the CMB polarization, modifications of thematter power spectrum, and the enhanced spindown of

rotating black holes. However, in order to give rise toobservable effects of this sort, the particle in questionmust be exceedingly light, with a mass m & 10�10 eV.In the extra-dimensional scenario we are discussing here,the Newton’s-law-modification constraint on the compac-tification scale Mc stated in Eq. (4.1) implies that all a� inthe tower have masses � * 10�3 eV in any scenario inwhich y & 1, i.e., in which the full tower of a� contributessignificantly to �tot. Consequently, the additional con-straints on ultralight axions and axionlike fields discussedin Ref. [9] are not relevant for our bulk-axion DDMmodel.

V. SYNTHESIS: COMBINEDPHENOMENOLOGICAL CONSTRAINTS ONAXION MODELS OF DYNAMICAL DARK

MATTER

In the previous section, we enumerated the individualastrophysical, phenomenological, and cosmological con-siderations which potentially constrain our bulk-axionDDM model, and we evaluated the restrictions that eachplaced on the parameter space of this model. In this sec-tion, we summarize how these individual results, takentogether, serve to constrain that parameter space. Ourparticular interest concerns the preferred region of parame-

ter space outlined in Ref. [2], namely fX � 1014–1015 GeV, �G � 102–105 GeV, and Mc chosen suffi-ciently small that y & 1. Indeed, this is the region withinwhich the full KK tower contributes nontrivially to the totaldark-matter relic abundance.In Fig. 12, we show the combined exclusion regions for a

purely photonic axion with c� ¼ 1 for �G ¼ 1 GeV (left

panel), �G ¼ 1 TeV (center panel), and �G ¼ 100 TeV(right panel). The shaded regions displayed in each of the

FIG. 11 (color online). Contours of the critical valueHcritI in ðfX;McÞ parameter space. As discussed in the text, choosingHI � Hcrit

I

guarantees that misalignment production dominates over vacuum fluctuations in determining the relic abundance ~�� of all a� in ourDDM ensemble, as desired. Here, we have taken TRH ¼ 5 MeV, Ne ¼ 100, and � ¼ gG ¼ ¼ 1 and assumed a photonic axion withc� ¼ 1. The left, center, and right panels display the results for �G ¼ 1 GeV, �G ¼ 1 TeV, and �G ¼ 100 TeV, respectively.


055013-39

plots are excluded by the various considerations discussedin Sect. IV. Specifically, the exclusion regions appearing inthese panels are those associated with helioscope limits onsolar axion production (red), collider considerations (ma-genta), tests of Newton’s-law modifications via Eotvos-type experiments (purple), measurements of the diffuseextragalactic X-ray and gamma-ray spectra (orange), ob-servations of the lifetimes of globular-cluster stars (yel-low), energy-loss limits from supernova SN1987A (cyan),the model-consistency requirement �G < fX discussed inRef. [2] (gray), and the 3 upper bound on the dark-matterrelic abundance from WMAP (brown). The additionalrequirement that the relic abundance be primarily deter-mined by the misalignment mechanism (as envisioned inour DDMmodel) excludes the green-shaded region, withinwhich a substantial population of a� is generated viainteractions with SM particles in the thermal bath. Theremaining unshaded regions of parameter space are theregions within which our DDM model is consistent withall of these constraints. The dashed black line indicates thecontour y ¼ �; smaller values of y correspond to theregion below and to the left of this line. As discussed inRef. [2], we are particularly interested in the unshadedregion of parameter space which falls below and to theleft of this line, since this is the region within which notonly are all of the aforementioned constraints satisfied, butalso the full tower of a� contributes nontrivially to �tot.

As we see in Fig. 12, for small �G the most stringentconstraint on the parameter space of our model is the onederived from energy-loss limits from SN1987A. The con-straint from globular-cluster stars is also reasonably strin-

gent, and the constraint derived from missing-energyprocesses such as pp ! �þ 6ET at the LHC is estimatedto be of roughly the same order. However, as the y ¼ �contour superimposed over each panel in Fig. 12 indicates,the full tower of a� contributes significantly to �tot for all�G * 100 GeV. Indeed, this is precisely the �G regimeassociated with the preferred region of parameter space forour model. We therefore conclude that within this region ofparameter space, a photonic bulk-axion DDM ensembleconstitutes a viable dark-matter candidate.In Fig. 13, we consider all of the same constraints as in

Fig. 12, but for the case of a hadronic axion with cg ¼c� ¼ 1. In this case, since the a� couple to hadrons, the

constraints from SN1987A and from axion production viainteractions among the SM particles in the radiation bathboth become even more stringent. Again, as in thephotonic-axion case, we find that the leading constraintfor small �G is that from SN1987A, and that as �G

increases, the model-consistency constraint becomes in-creasingly stringent. However, as in the photonic-axioncase, we see that within the preferred region of parameterspace for our model, a hadronic bulk axion is also consis-tent with experimental and observational limits. Thus ahadronic bulk-axion DDM ensemble is a viable dark-matter candidate as well.We also observe that the exclusion contours in Figs. 12

and 13 associated with SN1987A energy-loss limits,globular-cluster-star evolution, collider constraints, andaxion overproduction from SM particles in the radiationbath have the same slope. This is because all of theseconstraints involve the production of light axions which

FIG. 12 (color online). Exclusion regions associated with all applicable phenomenological constraints discussed in this paper for ourbulk-axion DDM model with �G ¼ 1 GeV (left panel), �G ¼ 1 TeV (center panel), and �G ¼ 100 TeV (right panel). In each case,we have taken � ¼ gG ¼ 1, TRH ¼ 5 MeV, andHI ¼ 10�3 GeV, and we have assumed that the axion only couples to the photon fieldwith c� ¼ 1. The shaded regions are, respectively, excluded by data from helioscope measurements (red), collider considerations

(magenta), tests of Newton’s-law modifications via Eotvos-type experiments (purple), measurements of the diffuse extragalactic X-rayand gamma-ray spectra (orange), observations of the lifetimes of globular-cluster stars (yellow), energy-loss limits from supernovaSN1987A (cyan), the model-consistency requirement �G < fX (gray), overproduction of thermal axions (green), and the upper boundon the dark-matter relic abundance from WMAP (brown). The dashed black line corresponds to y ¼ �; smaller values of y correspondto the region below and to the left of this line.


055013-40

are never directly detected, and thus involve physical pro-cesses whose amplitudes include a single coupling factorbetween the a� and a pair of SM fields. By contrast, theslopes of the constraint contours associated with otherclasses of physical processes can be quite different. Thehelioscope-constraint contour, for example, is related toprocesses in which axions are both produced and subse-quently detected via their interactions with SM fields.Likewise, the contour associated with limits on featuresin the diffuse X-ray and gamma-ray backgrounds is due toprocesses involving the decays of a preexisting cosmologi-cal population of axions, and therefore depends not only onthe couplings of the a� to SM fields, but to their relativeabundances as well. The slopes of these constraint contoursconsequently differ from those which characterize thecontours associated with SN1987A energy-loss limits,globular-cluster-star evolution, and so forth.

Finally, it is perhaps worth discussing the extent towhich the above results would be altered for higher valuesof TRH. Of course our primary motivation for adopting sucha low value for TRH in the first place is that the constraintsrelated to the production and decay of KK gravitons dis-cussed in Sec. IVA strongly prefer a reheating temperaturewithin the range 4 MeV & TRH & 20 MeV [7]. The pri-mary consequence of elevating TRH above this windowwould therefore be a conflict with these constraints.However, it is instructive to consider what effect increasingTRH would have on the DDM ensemble itself and thephysical processes which constrain it, independent of is-sues related to KK-graviton cosmology. Therefore, for thepurpose of this hypothetical discussion, we shall simplyassume that some method can be found for satisfactorilyaddressing these issues and proceed to address the impactof increasing TRH on dark-matter phenomenology.

The principal way in which elevating TRH affects thedark sector is via a modification of the cosmological

abundances of the a�. In particular, Eqs. (3.9) and (3.10),along with the time-temperature relation

tRH ¼ffiffiffiffiffiffiffiffiffi45

2�2

sg�1=2 ðTRHÞ MP

T2RH

; (5.1)

imply that ��ðtnowÞ / TRH for all a� which begin oscillat-ing at times t� & tRH. Thus, as long as the a� whichprovide the dominant contribution to�

tot begin oscillatingat such times, increasing TRH results in a proportionalincrease in �

tot. It therefore follows that our preferred

region of parameter space shifts to lower values of fX forfixed Mc and �G.The constraints on our model may also be affected by a

change in TRH. However, because a modification of TRH

only affects the cosmological context in which the a� areproduced and not the intrinsic properties of these particlesthemselves, such a modification will only have an impacton constraints which are sensitive to the cosmologicalabundances of the a�. Such constraints include the over-closure bound, as well as limits on features in the diffuseX-ray and gamma-ray backgrounds, on disturbances in theratios of light-element abundances, and on distortions inthe CMB. However, modifying TRH has little effect on theleading constraints on the parameter space of our model,which include collider constraints and energy-loss limitsfrom SN1987A. Indeed, the only leading constraint whichis affected by modifying TRH is the WMAP constraint on�

tot itself, discussed above.We therefore conclude that if TRH could be increased

without running afoul of the constraints from KK-gravitoncosmology, the primary result would be that the preferredregion of parameter space for our bulk-axion DDM model

would shift to lower values of fX for fixed Mc and �G.However, since the leading constraints on the model areessentially independent of the reheating temperature, we

FIG. 13 (color online). Same as Fig. 12, but for a hadronic axion—i.e., an axion which couples to both photons and gluons (andhence to pions, nucleons, and other hadrons), but not directly to SM quarks or leptons. For these panels, we have taken cg ¼ c� ¼ 1.

Note that the contour shown in the left panel corresponding to thermal overproduction (green) lies only slightly to the right of thecontour corresponding to collider constraints (magenta).


055013-41

find that an increase in TRH up to a few orders of magnitudecould easily be accommodated. However, any substantiallylarger increase in TRH would likely result in severe tensionsbetween relic-abundance constraints and other phenome-nological bounds.

VI. DISCUSSION AND CONCLUSIONS

In Ref. [1], we proposed a new framework for dark-matter physics which we call ‘‘dynamical dark matter.’’.The fundamental idea underpinning DDM is that the re-quirement of stability is replaced by a delicate balancingbetween lifetimes and cosmological abundances across avast ensemble of individual dark-matter components. InRef. [1], we developed the general theoretical features ofthis new framework. By contrast, in Ref. [2], we presenteda ‘‘proof of concept,’’ namely, an explicit realization of theDDM framework in which the DDM ensemble is realizedas the infinite tower of KK excitations of an axionlikefield propagating in the bulk of large extra spacetimedimensions.

In this paper, we have completed this study by system-atically investigating all of the experimental, astrophysical,and cosmological constraints which apply to this DDMmodel. Some of these constraints pertain to theories withlarge extra dimensions in general, while others pertainspecifically to our model. Among the bounds we haveconsidered are constraints from limits on a� productionby astrophysical sources such as stars and supernovae;constraints related to the effects of late relic-axion decayson BBN, the CMB, and the diffuse X-ray and gamma-raybackgrounds; collider constraints on missing-energy pro-cesses such as pp ! jþ 6ET and pp ! �þ 6ET ; con-straints on isocurvature perturbations generated as aconsequence of misalignment production; constraints onthe production of relativistic axions due to interactions inthe thermal bath after inflation; and constraints on thedirect detection of dark axions by microwave-cavity de-tectors and other, similar instruments. We have verified thatall of these constraints are satisfied within the preferredregion of parameter space for our model—namely, that inwhich the bulk-axion DDM ensemble accounts for theobserved dark-matter relic abundance, while at the sametime the full tower of axion modes contributes meaning-fully to that abundance. We therefore conclude that thisbulk-axion DDM model is indeed phenomenologicallyviable, and that the overall DDM framework is a self-consistent alternative to traditional approaches to thedark-matter problem.

While the focus of this paper has been on the specificbulk-axion DDM model presented in Ref. [2], we note thatmany of our results, and in many places our entire meth-

odology, have a far wider range of applicability. For ex-ample, much of the formalism developed in Sec. IV forevaluating the cosmological constraints on decaying darkmatter in our bulk-axion DDM model is applicable to anymodel in which the dark sector comprises a large numberof fields. This is true for issues as diverse as BBN, diffusephoton backgrounds, or stellar cooling. Likewise, irrespec-tive of issues pertaining to dark-matter physics, many ofour results and techniques may have applicability to theo-ries with large numbers of axions, such as the recentlydiscussed ‘‘axiverse’’ theories [9,68]. Thus, we believe thatthe methods developed and employed in this paper canserve as a prototype for future phenomenological studies ofnot only the DDM framework, but also, more generally,any theories in which there exist large numbers of interact-ing and decaying particles.We also note that while we have focused our attention in

this work primarily on the preferred region of parameterspace defined below Eq. (3.31), this is not the only regionwithin which our scenario constitutes a phenomenologi-cally viable model of dark matter. In particular, one of thecriteria which defines our preferred region is the property��Oð1Þ, since this condition implies that the full KKtower plays a nontrivial role in dark-matter phenomenol-ogy. Hence, this is the region which is the most interestingfrom a DDM perspective. However, there also exist sub-stantial regions of parameter space which are equallyphenomenologically viable (in the sense that �

tot ��CDM and all relevant constraints are satisfied), but withinwhich � � 0 and the lightest of the a� contributes essen-tially the entirety of the dark-matter relic abundance. Inthese regions, our DDMmodel reduces to the limiting caseof a traditional dark-matter model in which the lightestaxion mode plays the role of a standard dark-matter can-didate. Comprehensive reviews of the phenomenology as-sociated with generalized-axion dark-matter candidatesand the bounds on their masses, couplings, etc., can befound in Refs. [18,36,37,69].

ACKNOWLEDGMENTS

We would like to thank K. Abazajian, Z. Chacko, M.Drees, J. Feng, J. Kumar, R. Mohapatra, M. Ramsey-Musolf, S. Su, T. Tait, S.-H. H. Tye, X. Tata, and N.Weiner for discussions. K. R.D. is supported in part bythe U.S. Department of Energy under Grant No. DE-FG02-04ER-41298 and by the National Science Foundationthrough its employee IR/D program. B. T. is supported inpart by DOE Grant No. DE-FG02-04ER41291. The opin-ions and conclusions expressed herein are those of theauthors, and do not represent either the Department ofEnergy or the National Science Foundation.


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Phenomenological constraints on axion models of dynamical dark matter

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