Cosmological limits on axions and axion-like particles

axion-like parti les

axion-like parti les

Davide Cadamuro

Dissertation

an der Fakultät für Physik

der LudwigMaximiliansUniversität

Mün hen

vorgelegt von

Davide Cadamuro

from O tober 2009 until Mar h 2012 at the Max-Plan k-Institut für

Physik (Werner-Heisenberg-Institut), Mün hen, under the supervision

of Dr. Georg G. Raelt and Dr. Javier Redondo.

Erstguta hter: PD Dr. Georg Raelt

Das Axion ist einpseudo-Nambu-Goldstone Boson. Es tritt in Ers heinung na h

dem spontanen Bru h der Pe ei-Quinn Symmetrie, die als Lösung des starken

CP-Problems vorges hlagen wurde. Andere pseudo-Nambu-Goldstone Bosonen,

postuliertineinigenErweiterungendesStandardmodells,werdenAxion-Like

Par-ti les (ALPs)genannt, wenn siebestimmteEigens haften mitdem Axionteilen,

insbesonderedieKopplunganzweiPhotonen. BisjetztwarenalleSu hen na h

A-xionenundALPserfolglos. Diesbedeutet, dassderenKopplungenextrems hwa h

sein müssen. Allerdings können Axionen und ALPs einige beoba htbare

astro-physikalis he und kosmologis he Auswirkungenhaben, anhanddererman den

Pa-rameterraumdieser Teil hen eins hränken kann.

Wirkonzentrieren uns auf S hranken aus der Kosmologie,dieein ideales Feld

für dieUntersu hung vonAxionen und ALPs darstellt. Insbesondere untersu hen

wir als erstes die Mögli hkeit einer Axion- und ALP-Population, die während

der frühesten Augenbli ke des Universums entstanden ist. Die Bedeutung dieser

Analyse rührt daher, dass Axionen und ALPs wegen ihrer s hwa hen

We hsel-wirkung und der besonderen Produktionsme hanismen ideale Kandidatenfür die

dunkle Materie sind. S hlieÿli h betra hten wir die Folgen des Zerfalls dieser

Teil hen für bestimmte kosmologis he Observablen, nämli h für das

Photonen-spektrum von Galaxien, für den kosmos hen Mikrowellenhintergrund, für die

ef-fektive Zahl anNeutrinos und dieursprüngli he Häugkeit der Elemente. Unsere

S hrankenstellendenstriktestenTest einesfrühenZerfallsvonAxionenundALPs

ex-The axionis a pseudo-Nambu-Goldstone boson. It appears afterthe spontaneous

breaking of the Pe ei-Quinn symmetry, whi h was proposed to solve the

strong-CP problem. Other pseudo-Nambu-Goldstone bosons, postulated in some

ex-tensions of the standard model of parti le physi s, are alled axion-like parti les

(ALPs)iftheyshare ertain hara teristi swiththeaxion,inparti ulara oupling

to two photons. Thus far, axion and ALP sear hes have been unsu essful,

indi- atingthattheir ouplings havetobeextremelyweak. However, axions andALPs

ould be responsible for some observable ee ts in astrophysi s and osmology,

whi h an alsobe exploitedto onstrain the parameter spa e of these parti les.

Wefo usonlimits omingfrom osmology,whi hisanoptimaleld for

study-ing axions and ALPs. In parti ular, we rst investigate the possibility of a

pri-mordial population of axions and ALPs arising during the earliest epo hs of the

universe. Theimportan eofthisanalysisliesonthefa tthataxionsandALPsare

ideal darkmatter andidatesbe auseof their faintintera tionsand their pe uliar

produ tionme hanisms. Finally,we onsiderthe onsequen esofthede ayofsu h

a population on spe i osmologi al observables, namely the photon spe trum

of galaxies, the osmi mi rowave ba kground, the ee tive number of neutrino

spe ies, and the abundan e of primordial elements. Our bounds onstitute the

most stringent probes of early de ays and ex lude a part of the ALP parameter

Prefa e 1

1 Axions and axion-like parti les 5

1.1 The strong-CP problemand the axion . . . 5

1.2 Enlargingthe parameter spa e: axion-likeparti les . . . 13

1.3 A ompendiumof limits . . . 15

1.4 Where ould axions and ALPs hide? . . . 23

2 Establishing an axion or ALP reli population 25 2.1 Produ tionme hanisms I: non-thermalreli s . . . 28

2.1.1 Realignmentme hanism . . . 28

2.1.2 Before orafter ination? . . . 36

2.1.3 Contributionof topologi aldefe ts . . . 40

2.1.4 Dete tion of pseudos alar old dark matter . . . 41

2.2 Produ tionme hanisms II: thermal reli s . . . 43

2.3 Absorptionof overabundant pseudos alars . . . 50

2.4 Pseudos alar de ay . . . 52

3 Signals from the sky: reli de ay photons 57 3.1 Spe tral distortions of the osmi mi rowave ba kground . . . 58

3.2 Ionizationhistory of the universe . . . 62

3.3 Dire tdete tion of reli de ay photons . . . 65

4.1 Ee tivenumberof neutrinos . . . 75

4.1.1 ALP bounds . . . 77

4.1.2 Axionbounds . . . 83

4.2 Inuen e onbig-bang nu leosynthesis . . . 85

4.2.1 Axionsand BBN . . . 90

4.2.2 ALPsand BBN . . . 92

4.3 Ee t of other ouplings . . . 98

5 Summary and on lusion 103

A Axion-photon mixing 111

Dayafterday, osmologyhas be omemoreuseful asatoolforparti lephysi s. In

its earliest epo h, the universe provided the environment with the highestknown

energy,mu hhigherthanthosepresentlytestedin olliders. Westillmissaparti le

physi s des ription of su h high energy s ales, therefore many open osmologi al

questions are waiting for a satisfying answer in terms of mi rophysi s. Every

proposal of physi s beyond the standard model of parti le physi s (SM) must

deal with osmology, notablyprovidingme hanismsforinationandbaryogenesis

and andidates for dark matter and dark energy. All these issues provide further

motivationsto rea h a omplete pi tureof parti le theory.

Ontheotherhand,throughthestudyofthosephasesof osmi evolution,when

both the mi rophysi s and osmologi al des riptions are well settled, physi ists

an put severe onstraints on physi s beyond the SM. Big-bang nu leosynthesis

(BBN),the osmi mi rowaveba kground (CMB)and larges ale stru ture (LSS)

givepre iseindi ations about what happened whenthe universe wasolder thana

se ond. Moreover, in the last few years the vastly in reasing amount and quality

of data has propelled osmology toits pre isionera. New parti lephysi s models

have tofa e these broad datasets and must be onsistent with them. Cosmology

is thereforea ru ial testingground for parti lephysi s.

But osmologydoesnot playonlythekilljoyrole. Inparti ular,be auseof the

intrinsi longexposure timewhi hispe uliarof osmologi alphenomena,together

with the high luminosity provided by the universe itself, osmologygivesthe

pos-sibilityoftestingveryweaklyintera ting parti lesotherwise nota essibletohigh

energy parti le physi s experiments. Axion and axion-like parti les (ALPs) are

high-intensity astrophysi al settings like globular lusters or the sun. Cosmology

alsoplaysanimportantroleinthe quest for ndingthem,and, aswe willsee, the

bounds that itprovides are omplementarytothose provided by astrophysi s.

In the rst hapter, we introdu e the QCD axion and ALPs, providing some

theoreti almotivation. Wealsoreviewbriey thepresentstatusoftheexploration

of their parameter spa e through non- osmologi alapproa hes. Wealready

men-tionedthe astrophysi alobservations,andherewedes ribethe mainexperimental

approa hes to axionand ALP physi s. We on lude this hapter with some hints

for the presen e of su h parti les, supported by astrophysi al lues.

The primordialprodu tion of apopulationof axions and ALPsis the topi of

hapter2. Thisisthebasi questiontos rutinisebeforeeventostartthedis ussion

about the bounds osmology ould provide: without a reli populationthere an

be no osmologi al limits. Moreover, establishing a reli population of axions or

ALPsisanintriguingtopi initsown right,asit ouldprovidethe solutiontothe

dark matter mystery. The analysis of the stability of these parti les is therefore

the optimal way to on lude this hapter, and to introdu e the ore question of

this dissertation.

IfaxionsandALPsareunstable,theyde ayintophotons. Theirde ayprodu ts

ouldinuen e the evolution ofthe universe and the osmologi alobservables. In

hapter 3 we treat the limits oming from the late de ays of axions and ALPs.

These are obtained studying how the de ay ae ts the CMB and onsidering the

possibility of dire tly dete ting the photons emitted in the spe trum of galaxies

and the extragala ti photonba kground.

Atearliertimes,whentheuniversewashotanddenseenoughtorapidlylosethe

dire timprintofthe de ay produ tsviathermals attering, the de ay would have

amoresubtle andindire tinuen e. In hapter4we noti ethat the de ay,whi h

inje ts a large amount of photons in the primordial plasma, an ee tively dilute

baryons and neutrinos, if it happens after about ten millise onds sin e the big

bang. The primordialelementalabundan es and the amount of radiation during

the rst minutesof our universe are strongly ae tedby this dilutionand provide

further limits on the axion and ALP parameters. To on lude this hapter, we

our on lusions.

The resear h onaxions and ALPs will be one of the main frontiers of parti le

physi s in the near future, sin e these pseudos alars an solve some of the

unre-solved problemsof parti lephysi s. Their experimentaldis overywould be atrue

milestone along our path to understand Nature. The te hnologi al hallenge to

rea hthis aimispushingour apabilityofmeasuringextraordinarilysmallsignals,

but a long path remains to be overed. Astrophysi al and osmologi al

observa-tions provide dire tions about the parti le parameters, oering a guidan e in the

design of devoted dis overy experiments. The improving quality and quantity of

astrophysi aland osmologi aldatagivesus onden eonthepossibilitiesthesky

oers. That is why we onsider this kind of analysis of fundamental importan e

Axions and axion-like parti les

1.1 The strong-CP problem and the axion

Quantum hromodynami s(QCD),the theoryofstrongintera tions,in ludesaP,

T and thusCP violating term[47℄,

L

θ

= ¯

θ

α

s

8π

ǫ

µνρσ

2

G

ρσ

a

G

µν

a

,

(1.1) where

G

ρσ

a

is the gluon eld strength,

α

s

the ne stru ture onstant for olour intera tions and the so alled

θ

angle,

−π < ¯θ ≤ π

, is the ee tive parameter ontrollingCPviolation. Here, andinthefollowing,wewilluse naturalunitswith

~

_{= c = k}

_B

_{= 1}

, where

k

B

is the Boltzmann onstant. It is ommon pra ti e to writethe dualtensor

ǫ

µνρσ

G

ρσ

a

/2

simplyas

G

˜

aµν

, thuswewillsimplywrite

G ˜

G

for the tra e in the Lagrangian (1.1). The sour es of this

θ

-term are the Adler-Bell-Ja kiw anomaly of the axial urrent in QCD [8,9℄ and the topology of the QCD

va uum [10℄. There are no theoreti al hints about whi h value between

−π

and

+π

the parameter

θ

¯

ould hoose, so we justexpe t itto be an

O(1)

quantity. Thegluoneld

A

µ

a

transformslikea4-ve torunderC, P andT. We andene

the oloured ele tri and magneti elds as

E

_a

k

_{≡ G}

0k

_a

= ∂

0

A

k

_a

_{− ∂}

k

A

0

_a

+ g

s

f

abc

A

0

b

A

k

c

,

(1.2a)

B

_a

k

≡ ǫ

ijk

_G

ij a

= ǫ

ijk

∂

i

A

j a

− ∂

j

A

i a

+ g

s

f

abc

A

i b

A

j c

,

(1.2b)

and they transform like their ele tromagneti ounterparts under C, P and T. In

the

SU(3)

stru ture onstants. The olour indi es are

abc

, while

ijk

are spatial indi es. Be ause of the properties of the ompletely antisymmetri tensor

ǫ

µνρσ

, the Lagrangian (1.1) an be writtenas

G ˜

G =

−4 ~

B

a

· ~

E

a

. The s alar produ t of a polarve tor and anaxialve tor violatesP,Tand thusCP, on eCPT istaken for

granted.

WhetherCP isagoodsymmetryforQCDisnotafundamentalquastionbuta

phenomenologi alone. Atlowenergy the Lagrangian (1.1) indu esele tri dipole

moments in baryons, whi h have not yet been observed. In parti ular, many

measurements have been performed onthe neutron one. Cal ulations predi tthe

neutron ele tri dipole moment to be

d

n

∼ 10

−16

_{θ e cm}

_¯

[11,12℄, where

e

is the ele tron ele tri harge. Currently, the best experimental limit is

d

n

< 0.29

×

10

−25

_{e cm}

[13℄, whi htranslates into

|¯θ| . 10

−10

.

(1.3)

The essentials of the strong-CP problem are allhere: we were not expe ting su h

anextremely smallvalue for

θ

¯

.

Roberto Pe ei and Helen Quinn proposed an elegant solution to this

puz-zle, introdu ingaglobal hiral

U(1)

symmetry, alled the Pe ei-Quinn symmetry

U(1)

PQ

[14,15℄. Thissymmetryisspontaneouslybroken, anditsNambu-Goldstone boson(NGB)

a

,the elebratedaxion [16,17℄, ouplestogluonsbe ausethe

U(1)

PQ

symmetry is violated by the olour anomaly. Thus, another term involving

G ˜

G

enters in the QCDLagrangian,

L

aG ˜

G

=

−

a(x)

f

a

α

s

8π

G ˜

G ,

(1.4)

where

f

a

is the order parameter asso iated with the breaking of

U(1)

PQ

. Forthe momentwe ignorethe subtleties relatedtothe denition of

f

a

. At energies below the onnement s ale of olour intera tions,

Λ

QCD

≃ 1

GeV, gluons and quarks havetobeintegrated out,andQCDisdes ribed byanee tive hiralLagrangian.

Theterms (1.1)and (1.4),together withthe anomalous ontributionof the

U(1)

A

symmetry of the ee tive hiral QCD Lagrangian, be ome an ee tive potential

that an be parametrisedas

V (a)

_{≃ Λ}

4

_QCD



1

_{− cos}

 a

f

a

−

η

′

f

η

− ¯θ



.

(1.5)

Here,

η

′

isthepseudo-NGB(PNGB)oftheanomalous

U(1)

A

,and

f

η

isaparameter withdimensionsofamass. Theperiodi ityofthepotentialisduetothetopologi al

and instantoni nature of the

G ˜

G

Lagrangian. The onguration of minimum energy for the potential(1.5) is realised by the CP- onserving linear ombination

hai = ¯θ − hη

′

_i/f

π

f

a

, as required by the Vafa-Witten theorem [18℄. Sin e the

a

eld is massless, it an align itself with a null energy ost to the CP- onserving

point: thestrong-CPproblemissolved,independentlyofthevalueof

f

a

,

θ

¯

and

hη

′

_i

.

Be ause ofthe ee tivepotential(1.5), the mass of the eigenstate

η

′

_{+ a f}

η

/f

a

is

m

η

′

∼ Λ

2

QCD

/f

η

, whi h solves Weinberg's

U(1)

problem 1

. The orthogonal

om-bination, whi h mainly onsists of axion, mixes with the other s alar mesons in

the parti le spe trumand a quires mass [20℄. It isthis ombinationwhi his

usu-ally alled axion, and whose phenomenologyis studied. In rst approximation 

onsidering onlyup and down quark ontributions the axion mass is [21℄

m

a

=

m

π

f

π

f

a

√

_m

u

m

d

m

u

+ m

d

≃ 6 eV

 10

6

_GeV

f

a



,

(1.6)

where

m

π

= 135

MeV is the neutral pion mass and

f

π

= 92

MeV is the pion de ay onstant. The errors in the measurements of the light quark mass ratio

establish the range

m

u

/m

d

= 0.3



0.6

for this quantity [22℄, whi h translates into a 10% un ertainty on the axion mass,

m

a

= 5.23



6.01

eV for

f

a

= 10

6

GeV. The

preferred value

m

u

/m

d

= 0.56

gives us the result (1.6) after rounding it to 10% a ura y.

In the original Pe ei-Quinn model, the additional hiral symmetry was

im-posed on the SM through two Higgs doublets, linking the spontaneous breaking

of

U(1)

PQ

to the ele tro-weak symmetry breaking, thus

f

a

∼ E

EW

= 246

GeV. However, this rst attempt to solve the strong-CP problem through the

Pe ei-Quinn symmetry was qui kly ruled out, be ause the axion did not show up in

experimentaldata [23℄.

1

Anapproximate

U (1)

axial-ve tor urrentwouldrequirethepresen eofapseudos alarboson withmass smallerthan

√

3m

π

, whi h isnotobserved[19℄. This fa t was onsideredaproblem before realising that theaxial urrent isviolated by olour anomaly [8,9℄. Thusthe

θ

-term in the QCDLagrangiansolvesthe

U (1)

problem, and this provides signi an eto the

θ

-va uum. Of ourse, thepresen e ofthe axion isnot requiredto solvethe

U (1)

problem, but it helps to

Away tosavethe Pe ei-Quinnme hanism,bypassingtheexperimentallimits,

istoraisetheparameter

f

a

byseveralordersofmagnitude. Thes ale

f

a

suppresses boththe axionmassand ouplings,afa tthatwillbe omemore learshortly. For

themomentitissu ient tonoti ethat sin ethe axionisusually introdu edasa

phaseinthe Higgsse tor, itneedstobenormalisedbya onstant with dimension

of energy. It is therefore the dimensionless ratio

(a/f

a

)

that always appears in ee tive Lagrangians for axions. In reasing

f

a

automati ally lowers the axion ouplings to SM parti les, leaving untou hed the validity of the solution to the

strong-CP problem, as we have seen before. There are several implementations

of this idea, whi h are alled invisible axion models. The rst of them was the

Kim-Shifman-Vainshtein-Zakharov(KSVZ)axionmodel[24,25℄, whi h wedis uss

here in more detail for its simpli ity. A Dira quark eld

Q

 a olour triplet in the fundamental representation with no bare mass  and a SM gauge group

singletHiggs-s alar

S

are added to the standard model. The Lagrangian we need is

L

KSVZ

=

−

1

4

G

µν

a

G

aµν

+ ¯

θ

α

s

8π

G ˜

G + i ¯

Qγ

µ

_∂

µ

Q + g

s

A

µ

a

Qγ

¯

µ

λ

a

Q

− y



Q

†

_L

SQ

R

+ Q

†

R

S

∗

_Q

L



+

1

2

∂

µ

S

∗

_∂

µ

_S

− λ S

∗

_S

− v

2

a

2

,

(1.7) where

γ

µ

are the Dira matri es,

g

s

is the strong intera tion oupling onstant,

A

µ

a

thegluoneld,

λ

a

are theGell-Mannmatri es,

Q

L(R)

isthe left(right)handed proje tion of the Dira spinor,

y

the Yukawa oupling,

λ > 0

and

v

a

≫ E

EW

are the parameters of the Mexi an hat potential,the last one with dimensions of

energy. Forthemomentwedonottakeintoa ountthe intera tionof

Q

withthe ele troweak gauge elds. Under the global hiral

U(1)

PQ

, the elds transform as

Q

L

→ Q

L

e

iα/2

(1.8a)

Q

R

→ Q

R

e

−iα/2

(1.8b)

S

_{→ Se}

iα

,

(1.8 )

leavingthe Lagrangian

L

KSVZ

invariantat the lassi allevel.

It is onvenient toexpress the s alar Higgs singletin itspolarform,

S(x) = ρ(x) exp



i

a(x)

v

a



.

(1.9)

a

G

G

Q

Q

Q

Figure 1.1: Triangle loop diagram for the ee tive axion-gluon intera tion of

equation(1.12) .

At energies lower than

v

a

,

S

rolls toward the minimum of the Mexi an hat po-tential, and we an make the substitutions

ρ

→ v

a

. Ifwe limitour physi s onsi-derations toenergies lowerthanthe singletmass, we ankeep onlythe phaseeld

a

. This eld hanges as

a

→ a + αv

a

under PQ transformations (1.8). However, be ause of the invarian e of the Lagrangian (1.7), the energy ost of this shift

is null, and therefore

a

is the NGB for

U(1)

PQ

. Under this approximation, the Lagrangian is now

L

′

KSVZ

=

−

1

4

G

µν

a

G

aµν

+ ¯

θ

α

s

8π

G ˜

G + i ¯

Qγ

µ

_∂

µ

Q + g

s

G

µ

a

Qγ

¯

µ

λ

a

Q

− yv

a



Q

†

_L

e

i

va

a

Q

_R

+ Q

†

R

e

−i

a

va

Q

_L



+

1

2

∂

µ

a∂

µ

_{a .}

(1.10)

The spontaneous breaking of the Pe ei-Quinn symmetry produ es an ee tive

mass term

m

Q

= yv

a

for the quark eld, on e the phase terms between the left and right spinorsare removed. To a hieve this, we hiral rotate the quark eld,

Q

L

→ Q

L

exp



i

a

2v

a



,

Q

R

→ Q

R

exp



−i

_2v

a

a



.

(1.11)

This transformation adds tothe Lagrangian(1.10) the terms

δ

_{L = −}

∂

µ

a

2v

a

¯

Qγ

5

γ

µ

Q

−

α

s

8π

a

v

a

G ˜

G ,

(1.12)

wherethelasttermisthe ontributionof the olouranomaly,originatingfromthe

are

L

KSVZ

=

−

1

4

G

µν

a

G

aµν

+



¯

θ

₋

a

v

a

 α

s

8π

G ˜

G +

1

2

∂

µ

a∂

µ

_{a ,}

(1.13)

amongwhi hthere is the

L

aG ˜

G

term of equation(1.4).

If many heavy quark elds

Q

are introdu ed in this model, ea h of them would produ e a ontribution like equation (1.12). Under a PQ-transformation,

these heavy quarks transform if a PQ harge

q

i

PQ

is assigned to them, i.e.

Q

j

L

→

Q

j

_L

exp(iq

_PQ

j

α/2)

and

Q

j

R

→ Q

j

R

exp(

−iq

j

PQ

α/2)

if

S

→ Se

iα

. This leaves the

La-grangian invariant under the ondition

q

j

PQ

= 1

. In the ee tive Lagrangian we dene the Pe ei-Quinn s ale oraxion de ay onstant to be

f

a

≡

v

a

N

,

(1.14) where

N =

P

j

q

j

PQ

ounts the number of spe ies that are PQ- harged, and the

formof

L

aG ˜

G

of equation(1.4) is restored.

A triangle diagram,like that of gure 1.1, an reate other anomalous

ontri-butionstoequation(1.12),iftheheavyquarkeldsarealso oupledtoele troweak

gaugebosons. In the low-energyregime, this means the presen e of a two-photon

ouplingfor the axion,

L

aγγ

=

−

g

aγγ

4

aF ˜

F ,

(1.15)

F

µν

being the photoneld strength and

F

˜

µν

its dual. Anyway, be ause of axion-meson mixing, the two-photon oupling arises even if the heavy quark eld is

ompletelyde oupled fromthe

SU(2)

× U(1)

gaugese tor. The oupling onstant

g

aγγ

is a tually the sum of two ontributions [5℄,

g

aγγ

=

α

2πf

a

1

N

N

X

i=1

q

_PQ

i

q

_EM

i

2

−

2

3

4m

d

+ m

u

m

d

+ m

u

!

≡

α

2πf

a

C

γ

,

(1.16)

where

α

isthenestru ture onstant. Thersttermistheele tromagneti anoma-lous ontribution of the heavy quarks,

q

i

PQ (EM)

being the Pe ei-Quinn

(ele tro-magneti ) harge ofthe

i

-th

Q

eld,and these ondone isprodu edby the axion-mesonmixing. Thismodeldependen e anbeparametrisedwiththedimensionless

quantity

C

γ

. In the literature, the KSVZ axionisusually dened tohaveno ele -tromagneti anomaly and thus tohave

|C

γ

| ≃ 1.95

for

m

u

/m

d

= 0.56

. Be ause of

theun ertainupanddownquarkmassratio,theparameter

|C

γ

|

ouldbeanywhere between 1.92 and 2.20.

Another popular axion model was proposed by Zhitnitsky [26℄ and by Dine,

Fis hler and Sredni ki [27℄ and is alled the DFSZ axion. Again, the axion is

relatedto the phaseof a Higgs-likes alarsinglet

S

whi his addedtoa two Higgs doublet extension of the SM. The s alar

S

is oupled only to the Higgs doublets

H

u

and

H

d

in the Higgs potential,

V (H

u

, H

d

, S) = λ

u

H

u

†

H

u

− v

u

2

2

+ λ

d

H

d

†

H

d

− v

d

2

2

+ λ S

∗

_S

− v

2

a

2

+ χ H

u

†

H

u

H

d

†

H

d

+ ζ H

u

†

H

d

H

d

†

H

u

+

γ

u

H

u

†

H

u

+ γ

d

H

d

†

H

d

 S

∗

S + γ



H

u

†

H

d

S

2

+ h.c.



,

(1.17)

where

λ

u

,

λ

d

,

λ

,

χ

,

ζ

,

γ

u

,

γ

d

and

γ

are real dimensionless parameters of the potential. This potentialis invariant underthe PQ-transformation

H

u

→ H

u

e

iX

u

,

H

d

→ H

d

e

iX

d

,

S

→ Se

i(X

u

−X

d

)/2

.

(1.18)

Themodel anequallyworkifthelasttermofthepotentialis

γ



H

†

u

H

d

S + h.c.

, but this time

γ

has tohave mass dimensionsand the PQ-transformations have to be adapted. The Yukawa Lagrangian is

L

Y

= y

u

ǫ

ab

Q

¯

a

L

H

u

†b

u

R

+ y

d

Q

¯

L

H

d

d

R

+ y

d

L

¯

L

H

d

l

R

+ h.c. ,

(1.19)

where

ǫ

is the

2

× 2

antisymmetri matrix and, in this ase,

Q

L

and

L

L

are the left-handed

SU(2)

quark and lepton doublets of the SM. The right-handed hargedlepton omponentis

l

R

,andtheYukawa ouplingsarethe

y

s. Theva uum expe tation values (VEV) of the Higgselds an bewritten as

hH

u

i =

1

√

2

0

v

u

!

,

hH

d

i =

1

√

2

0

v

d

!

,

hSi = v

a

.

(1.20)

The two Higgs doublets have four degrees of freedom ea h, and three of them

are

SU(2)

phases, while

S

has got just a radial ex itation and a phase, like in the KSVZ model. For phenomenologi al reasons we want

v

a

≫ v =

pv

2

u

+ v

2

d

∼

the two Higgs doublets an be written in the unitary gauge deleting from the

theorythe redundantmasslessdegrees of freedom,whi hare eaten by the gauge

bosons. At this point, the neutral orthogonal ombination of degrees of freedom

whi h survives mixes with the phase of the

S

boson due to the last term of the potential(1.17). Inthismixing,one ombinationismassiveandistheHiggsboson,

whilethe orthogonalmassless NGB is the axion

a

. Redening the Higgsdoublets through transformations like (1.18) to reabsorb the axion phase in the last term

of the potential (1.17), makes

a

reappear in the Yukawa terms (1.20). Again, the axion phase an be reabsorbed with some hiral rotations of the fermioni

elds, produ ing a set of anomalous terms like equations (1.12) and (1.15), but

thistimewiththeSMfermionsinsteadoftheheavyquark

Q

. IntheDFSZ model, the derivative ouplings in (1.12) play a role inthe lowenergy theory, sin e they

involve the light fermions too. The strengths of the ouplings depend on the PQ

harges

X

u

and

X

d

, and are therefore modeldependent.

We presently do not know whi h are the features of the high energy model

whi hgivesorigintothe axion. However, we an adheretotheproposalofGeorgi,

Kaplan and Randall [28℄, and write a generi low energy ee tive theory. In the

rangebetween

E

EW

and

Λ

QCD

,the Lagrangian is

L

a

=

1

2

∂

µ

a∂

µ

_a

−

a

f

a

α

s

8π

G ˜

G

−

g

aγγ

4

aF ˜

F

−

∂

µ

a

2f

a

X

f

C

f

ψ

¯

f

γ

5

γ

µ

ψ

f

.

(1.21)

All the model dependen ies are hidden in the oupling oe ients

g

aγγ

and

C

f

, where the index

f

runs over all the SM fermions. The denition of the axion-photon oupling onstant

g

aγγ

is provided by equation (1.16). In parti ular, the KSVZ modelpredi ts

C

f

= 0

for all the leptons and the ordinary quarks at tree level. In the DFSZ model, the oupling oe ienttoele trons is

C

e

= cos

2

_{(β) /3}

,

while the ouplings to the up and down quarks are respe tively

C

u

= sin

2

_{(β) /3}

and

C

d

= cos

2

_{(β) /3}

,where

β

is the ratioof theva uumexpe tationvalues ofthe two

H

u

and

H

d

elds [22℄. Below

Λ

QCD

, gluons and quarks onne sowe have to write anee tive Lagrangian in ludingthe ouplings to nu leonsand mesons,

L

a

=

1

2

∂

µ

a∂

µ

_a

− m

2

a

f

a

2



1

_{− cos}

 a

f

a



−

g

aγγ

4

aF ˜

F

−

∂

µ

a

2f

a

X

f

C

f

ψ

¯

f

γ

5

γ

µ

ψ

f

+

L

aπ

.

The potentialis the same of equation (1.5) up toa reparametrisation. This time,

the sum over

f

overs the light SM leptons and the nu leons. The oupling o-e ients for axion-proton and axion-neutron intera tions are

C

p

=

−0.55

and

C

n

= 0.14

for

m

u

/m

d

= 0.3

and

C

p

=

−0.37

and

C

n

=

−0.05

for

m

u

/m

d

= 0.6

in the KSVZ model [22℄. The DFSZ axion has ouplings to nu leons of the same

order whi h depend alsoon

β

[22℄. Among the intera tionswith mesons, onlythe pion-axionone isinteresting for our purposes. Therefore in Lagrangran (1.22)we

express

L

aπ

as[22℄

L

aπ

=

∂

µ

a

f

a

C

π

f

π

π

0

π

+

∂

µ

π

−

_{+ π}

0

_π

−

_∂

µ

_π

+

− 2π

+

π

−

_∂

µ

_π

0

,

(1.23)

where

C

π

isagain a modeldependent onstant.

1.2 Enlarging the parameter spa e:

axion-like parti les

As explainedinthe previousse tion,several axionmodelsexist,ea h ofthem

sol-vingthestrongCP problem,butprovidingdierent ouplingstotheSMparti les.

Moreover, the PQ-me hanism works for every value of

f

a

, and the lues about the hara teristi PQ-s ale are only spe ulative. The experimentalsear h for the

axion requires therefore not only to explore many orders of magnitude in

f

a

or

m

a

, but even to s an the model dependen ies held in the oupling oe ients of Lagrangians (1.21) and (1.22).

AftertheseminalpaperbySikivie[29℄, themost relevantaxiondire tsear hes

try to exploit the two-photon oupling. However, as we will show in the next

se tion, the dis overy task is tough, the axion being very weekly oupled for the

allowed range of

f

a

. Anyway, this kind of experiments ould nd in prin ipleany kind of pseudos alarparti le

φ

oupledto photons through a term like

L = −

1

4

g

φ

φF

µν

F

˜

µν

_,

(1.24)

lastyears,andthenameaxion-likeparti le(ALP)hasbeen oinedforthem. These

ALPs an in prin iplealso ouple to other parti les besides photons, however we

willalwaysdeal with the intera tion Lagrangian (1.24),as itistypi ally the most

important one for the phenomenology at low energies. We will omment about

other possible ouplings atthe end of hapter4.

ALPsare even moreinteresting onthe theoreti alside,asthey anarise inthe

lowenergyspe trumofmanyextensionsoftheSM.AnALP anappearinatheory

as a PNGB of a ontinuous global symmetry. Examples of these symmetries are

relatedtoparti leavour [32℄,leptonnumber[33,34℄orthe

R

-symmetryin super-symmetry [35,36℄. When a ontinuous global symmetry is spontaneously broken,

masslessNGBsappearinthelowenergytheoryasphasesofthehighenergydegrees

of freedom. Sin e phases are dimensionless, the anoni allynormalised theory at

low energies always involvesthe ombination

φ/f

φ

, where

φ

is the NGB eld and

f

φ

is a s ale lose to the spontaneous symmetry breaking (SSB) s ale. The ALP ould be for instan e related to the generation of right-handed neutrino masses,

andhen ehaveade ay onstantatanintermediates ale,like

f

φ

∼ 10

10



10

12

GeV;

alternatively it ould be asso iated with a grand-uni ation theory (GUT), and

have a de ay onstant at the orresponding s ale

f

φ

∼ 10

15

GeV. From the

no-hairtheorem, weknowuptosomeextentthat bla k-hole dynami sviolatesglobal

symmetry onservation. Therefore unbroken global symmetries an not exist in

theories with gravity, and we should have PNGBs instead of NGBs. There are

manypossibilitiesfor breakingthe shift symmetry besides gravityee ts,

expli i-tly orspontaneously, perturbatively or non-perturbatively.

Moreover, theobservation thatinstringtheory ALPsappearinall

ompa ti- ationshas raised even more attention to them[37℄. These so- alledstringaxions

share the NGB properties (having ashift symmetry and being periodi )but with

the naturalsize of

f

φ

being the string s ale.

The fa t that the ALP a quires a mass implies that in the model Lagrangian

a potential has to be in luded. Taking inspiration from the axion ase, the ALP

potential an typi ally be parametrizedas

V (φ) = m

2

_φ

f

_φ

2



1

− cos

 φ

f

φ



.

(1.25)

a hara teristi s ale

Λ

,theALPmassisparametri allysmall,sin eitissuppressed by powers of

Λ/f

φ

. The phenomenology of the SM requires

Λ

to be related to physi s beyond the ele troweak s ale, i.e.

Λ &

TeV (whi h implies

m

φ

≫ m

a

for

f

φ

= f

a

), or to belong to a hidden se tor. We have here no pre on eptions regarding ALP mass, for it depends on the unknown ratio of two un onstrained

energy s ales.

Thedimensionful ouplingparameter

g

φ

inequation(1.24) anbeparametrised as

g

φ

≡

α

2π

C

γ

f

φ

.

(1.26)

In the simplest ase

C

γ

is an integer, but this is not true in general when the ALP mixes, either kineti ally or via symmetry-breaking ee ts with other ALPs

orwith pseudos alarmesons. Forstringaxions, the ouplingto photonsisrelated

via aloopfa tor to eitherthe string s ale or the Plan k s ale,or it ould beeven

weaker, soanALP with alarge ouplingwould restri tthe string s ale tobelow.

For string and eld theoreti al models the most interesting values are therefore

g

φ

∼ 10

−11



10

−15

GeV

−1

,

∼ 10

−19

GeV

−1

and

∼ 10

−21

GeV

−1

orresponding to

intermediate, GUTor Plan k s ales.

Wewilladheretothephenomenologi alapproa htoleavetheALPparameters

m

φ

and

g

φ

free to span many orders of magnitude, exploring the onsequen es of the presen e of these parti les in order to limit the ALP parameter spa e, or to

nd hints of their existen e.

1.3 A ompendium of limits

The sear h for axions and ALPs has not yet been su essful. Up to nowwe have

only indi ations about where these s alars are not and some hints about where

they ouldhide. The present bounds onthe axionmass, and onsequently on the

PQ-s ale, are plotted in gure 1.2. These bounds are indi ative, asthe ouplings

an hange a ordingtothe axionmodel, but they giveapi tureof the situation,

espe ially onsidering

O(1)

oupling oe ients. The red and brown bounds in the rst line, labelled Cold DM, Topologi al defe t de ay and Hot DM are

Hot DM

Topological

defect decay

Cold DM

ADMX

IAXO

CAST

Te

lescope

Laboratory

HB Stars HphotonsL

HB Stars HelectronsL

SN 1987A

Burst duration

Kamioka

16

O

White dwarfs

cooling

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Log

₁₀

H

m

a



eVL

Log

₁₀

H

f

a



GeVL

Figure1.2: Axionlimits. Theredboundsintherstline omefrom osmologi al

onsiderations. Theblueonesinthese ondlinearedue todire t measurements.

The green limits refer to astrophysi al arguments related to stellar evolution.

Theseboundsaredes ribedinmoredetailinthetext. Thedottedellipsesuggest

thevalues of

m

a

that ouldhelptting re ent white dwarf ooling data.

hapters. The blue bounds in the se ond line ome from dire t measurements of

astrophysi al and osmologi al quantities and from experiments, while the green

ones are related toastrophysi s.

As we have already mentioned,very soon after itwas proposed the axion was

ruled out in the keVMeV mass range. Measurements of heavy quarkonium state

de ays or of nu lear de-ex itations and beam dump or rea tor experiments have

foundnoeviden eforanaxion oupledtofermionsornu leonsupto

f

a

&

10

4

GeV,

whi h in term of mass is

m

a

.

0.6

keV [7℄. Pure parti le physi s experiments an only test values

f

a

in a rather low energy range. The bound they put dies out when

f

a

is high enough to fade the produ tion and dete tion probabilities throughthesuppressionof theaxion ouplings. Ingure1.2this boundislabelled

Laboratory.

The Teles ope region,

m

a

= 3



27

eV and

f

a

= 2.3

× 10

5



2.1

× 10

6

GeV, is

ex luded by the non-observation of photons that ould be relatedto the reli

ax-ionde ay

a

→ γγ

inthe spe trumofgalaxies andinthe extragala ti ba kground light [3841℄. The Axion Dark Matter eXperiment (ADMX) has provided some

onstraintsinthe region

m

a

= 1.9



3.5 µ

eV,whi htranslates into

f

a

= 1.8

× 10

12

3.3

×10

12

GeV,afterthe sensitivitytotesttheaxionwasalmostrea hed[42℄. This

ollaborationis looking for axion dark matter (DM) using an halos ope. We will

des ribe this instrument inthe next hapter. The plans of the ADMX

ollabora-tion are to s an the mass range en losed in the dashed region alled ADMX in

gure 1.2 aftersome upgrades [4345℄. Thetwolatter limitsare losely related to

osmology and to the axionbeing a DM omponent, we willtherefore referagain

to them inthe following hapters.

Stars represent a proli environment for the produ tion of light and

weakly- oupledparti les,aswehavelearnedforexampleintheneutrino ase[46℄. Thesun

issurely thebrightestaxionsour einthesky. A photon an onvert intoanaxion

ifitintera tswithanexternalmagneti orele tri eldbymeansofthetwo-photon

oupling of equation (1.15) as in gure 1.3 [47℄. This is the so alled Primako

ee t, whi h was rst proposed for the reation of mesons in the ele tri eld of

nu lei [48℄. Therefore, the high on entration of thermal photons, together with

the strong ele tromagneti elds of the stellar plasma, makes the sun, and stars

in general, ari hsoil for axionand ALP produ tion. The total axion luminosity,

al ulated using the standard solar model, is

L

a

∼ (g

aγγ

/10

−10

_GeV

−1

₎

2

₁₀

−3

_L

⊙

,

where

L

⊙

= 3.90

×10

25

Wisthesolarluminosityinphotons[49℄. Todete ttheux

of axions, several solar axion teles opes, like SUMICO [50℄ and CAST [51℄, have

been built. These helios opes are essentially va uum pipes. They are permeated

by astrongmagneti eldtoexploitthe inverse Primakoee t to onvert axions

ba k intophotons [29℄. Axions an enter the teles opebe ause of their very weak

intera tionwithmatter,andsu essivelybedete tedon etheyhaveos illatedinto

photons. Among the helios opes, CAST urrently givesthe strongest onstraints:

its results ex lude the part of the axion parameter spa e,

m

a

= 0.39



0.64

eV and

f

a

= 9.8

× 10

6



1.6

× 10

7

GeV, labelled CAST. Re ently, a new proposal for an

axionhelios opehasappeared,the InternationalAXionObservatory (IAXO)[52℄.

The hope is to improve the sensitivity to

g

aγγ

of at least one order of magnitude with respe t to CAST and therefore, in the most optimisti s enario, to explore

the area labelledIAXO whi h isen losed by the blue dashed line.

Waiting for IAXO and its results, the best upper limitson the axion mass in

gravi-a

g

aγγ

γ

γ

~

B

Figure 1.3: Diagram of thePrimako ee t.

approximately the same age and they dier only in their initialmass. Sin e the

moremassiveastar is,the faster itevolves, aglobular lustergivesthe possibility

tostudy a broadsample ofstellar evolution stages andto estimatehowlong ea h

phase lasts. In parti ular, if axions are produ ed inside a star and es ape, they

provide an additional ooling hannel, besides the photon and neutrino ones. If

there are more e ient energy release hannels, the nu learfuel onsumptionhas

to be faster, and thus the ageing qui ker. Counting the stars in ea h evolution

stage inside a globular luster permits us to study how fast the fuel onsumption

is and therefore to put bounds on the produ tion of axions in stellar ores. The

best onstraints omefromthestarswhi hhaverea hedtheheliumburningphase,

whi hare alledhorizontalbran h (HB)stars be auseof thepositionthey o upy

in the Hertzsprung-Russel diagram. The non-standard energy loss prolongs the

redgiant(RG)phaseandshortens the HBone [46℄. Countingthe RGand theHB

starsinglobular lustersand omparingthetwonumbers itispossibletoevaluate

the axion produ tion rate in stars, and to obtain the two HB Stars bounds in

gure 1.2. In parti ular, if the axion is dire tly oupled to the ele tron, i.e.

C

e

is

O(1)

,ithasasigni antprodu tion hannelmorewhi hisree tedinthebroader ex lusionbound. HBstarshaveatypi al oretemperatureof

T

∼ 10

8

K

∼ 10

keV. The thermal distribution of photons, averaged over the large volume of the star,

still in ludes many

γ

s that are energeti enough to e iently produ e axions if their mass isnot

m

a

&

300

keV, whi h iswhere the HB bounds stop.

Also supernova explosions (SN) are used to put limits on axions. Stars with

68

M

⊙

mass or more rea h the ultimatephase of the pro essing of nu lear fuel, reating an iron nu leus. Iron has the largest binding energy per nu leon and

the radiationpressure ne essary to ontrast thegravitationalpulland tomaintain

the hydrostati equilibrium. If the iron ore rea hes a riti al mass, it ollapses

under its own weight. On e the nu lear density is rea hed the ollapse stops and

the boun e produ es a sho k wave that expels the outer layers in a ore- ollapse

SN explosion. In the ollapse,ele trons are jammedinside protonsfor ing inverse

β

-de ays

e

−

_{+ p}

+

_{→ n + ν}

e

: aneutronstar formsandlotsof neutrinosare reated.

The density ofmatter inaSN ore issohigh,that even neutrinos remaintrapped

andittakessometimebeforethey andiuseout[46℄. AfterSN1987A,24neutrino

events were measured above the ba kground in about 10 s. Their distribution in

energy and time agrees wellwith the standard pi turefor typeII SNe. A parti le

with aweakermatterintera tionthan theneutrino would provideamoree ient

energydissipation hannelthanthestandard ones. Ifthis isthe ase,the neutrino

burst duration would have been shorter than what was measured. In su h a high

nu leondensityenvironment,axionswould beprodu edby virtueof theirnu leon

ouplinginrea tions like

N + N

→ N + N + a

. In gure1.2, the upperbound on the axionmass of SN 1987A,

m

a

< 16

meV,thus

f

a

> 4

× 10

8

GeV, omes from

this burst duration argument. Lighter axions, and thus less oupled ones, would

not bee ientlyprodu edand thusthey wouldhavenot signi antlyae ted the

timingofSN 1987Aneutrino events[49℄. On theother side,if axions ouplemu h

more strongly to matter, they ould be trapped inside the ore, and the neutrino

burst would have suered little or no modi ation. This is why the SN 1987A

boundstops atlow

f

a

. However, inthis ase someaxionsareemittedanditwould havebeen possible todete t theminthe KamiokandeII experimentthanks tothe

nu lear rea tion

a +

16

_O

_→

16

_O

∗

_→

16

_{O + γ}

. The region labelled Kamioka

16

O is

ex luded by the non-observation of these events [53℄. The SN 1987A bounds are

very un ertain and have to be taken with a grain of salt. They are parti ularly

interesting iftheaxionhas notadire t ouplingwiththeele tron, forthe stronger

limits oming fromHBstars are not validin this ase.

Mostof the bounds justdes ribed are also validin the ALP ase, as they are

dire tly onstraining

g

aγγ

and thus

g

φ

. Moreover, those based on measurements of the ouplingtoele trons ornu leons an alsobeused to onstrain ALPs, on e

e

+

_+e

-

_{® Γ+inv.}

U® Γ+inv.

Beam

dump

ALPS

PVLAS

CAST+SUMICO

HB

SN

SN Γ burst

Haloscopes

Cosmology

Axion

models

MWD

White dwarf

cooling

Transparency of the universe

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

-16

-14

-12

-10

-8

-6

-4

-2

Log

₁₀

H

m

Φ



eVL

Log

10

Hg

Φ

GeV

L

Figure 1.4: ALP limits. The blue regions are onstrained by parti le

physi s experiments. In green are the astrophysi s related limits, in parti ular

CAST+SUMICO are helios ope measurements. The laser experiments ex lude

theyellowregion. Halos opeshave exploredthe brown regionand inredarethe

boundsprovided by osmology. The dottedellipses suggestthevaluesof the

pa-rameterspa e whi hshould hara terisea pseudos alar parti leinorderto solve

someastrophysi al onundrums. Thebandofaxionmodels in ludesvalues of

C

γ

between 0.6and 6.

more onstrainingexperimentsandobservationsthantheaxion. Figure1.4,where

we olle tedthemostrelevantALPbounds,providesapanorami viewoftheALP

parameter spa e,in omparison with the bla k-dashed band whi h represents the

axionmodels, whose entralvalue is the KSVZ axion and the shading overs the

interval

0.6 . C

γ

.

6

.

sear h for ALP DM with ADMX and other halos opes. In blue are the bounds

omingfromparti lephysi sexperiments,ingreenastrophysi alobservationsand

arguments. The yellow area is onstrained by laser experiments dedi ated to the

sear hfor ALPs.

Parti le physi s experiments,as inthe axion ase, an set onstraints onlyfor

quite strongly oupled parti les and they pile up in the upper part of gure 1.4.

They are very useful anyway, for they an test larger mass than astrophysi s.

In gure 1.4 we draw in blue the ex lusion bounds due to

Υ

de ay and positro-nium annihilationintoinvisible hannels, labelled respe tively

Υ → γ + inv

and

e

+

e

−

_{→ γ + inv}

. The beam dumpexperimentsperformedin SLACex lude the

pat hlabelled Beam dump [54,55℄.

Astrophysi splaysa entralrole in onstrainingthe ALP ase. TheHBbound

on

g

aγγ

des ribed before is dire tly appli able to ALP photon oupling, and it is plotted in gure 1.4 in light green and labelled HB. The SN 1987A gives some

onstraints too. The rst one, labelled SN, is given by the duration of the

mea-sured neutrino pulse, as in the axion ase, and was derived in [30℄. The se ond

bound relatedto SN1987A is labelledSN

γ

burst and itis relatedtothe trans-paren y of the dense SN ore to the ALP propagation. If an ALP exists in this

region of the parameter spa e, it would be produ edduring the ore ollapse and

it would subsequently es ape from it. Then, the propagating ALPs an os illate

into high energy photons intera ting with the gala ti magneti eld, and nally

dete ted on earth. Sin e no

γ

-ray pulse was measured in orresponden e of SN 1987A, the SN

γ

burst region an be ex luded [56℄. However the SN bounds an be onsidered rather weak, as they rely on an insu ient understanding of

the SN dynami s and, in the ases just dis ussed, of the ALP emission from a

nu lear-density environment.

The CAST helios ope onstrains a large part of ALP parameter spa e. The

Japanese experimentSUMICOgivesalsoaboundinasmallpartoftheparameter

spa e not onstrained by CAST [50℄. Their limits are both plotted in dull green

in gure 1.4 and labelled CAST+SUMICO. It is interesting to noti e in this

pi ture how it is a tually di ult to rea h the sensitivity to onstrain the axion.

A lightshining through the wall (LSW) experimentis the table-top version of

an helios ope, in the sense that in this ase the sour e is not the sun but a high

intensity laser. The experiment onsists of a pipe divided in two by an opaque

barrier, the wall, in a produ tion se tion and a dete tion one. In both of them a

high va uum has been reated and an intense magneti eld imposed. The beam

of photons emitted by the laser passes through the magneti eld before ending

its run against the wall, onverting some of the photons of the beam into ALPs

via the Primako ee t. Be ause of their feeble oupling, ALPs an ross the

wallpassingintothe dete tion se tion,wherethey have tobe onverted ba k into

photonsintera tingwith themagneti eldbeforebeing nallydete ted [57℄. The

yellowpat hlabelledALPSingure1.4shows the onstraintsofthe latestLSW,

the Any Light Parti le Sear h(ALPS) [58℄.

Measuring hangesin polarizationis a dierent approa hto laser experiments

[47,59℄. Again, the laser beam passes through an intense magneti eld. The

omponent of the ve tor potential parallel to the magneti eld ee tively mixes

with the pseudos alar eld and itis partially absorbed and retarded. This auses

alightbeam,linearly polarisedat agiven anglewith the magneti eld, torotate

a small amount be ause of the absorptive pro ess and to gain a small ellipti ity

due to the dispersion ee t. Therefore, if pseudos alar parti les oupled to two

photons exist, a magneti eld indu es respe tively di hroism and birefringen e

on the magnetised va uum. Both ee ts are proportional to the square of the

magneti eld and of thephoton oupling

g

φ

. Moreover, the a quiredellipti ityis proportional to

m

2

φ

[60℄. The QED ba kground pro essesare virtualpair reation

forbirefringen e[61℄andphotonsplittingfordi hroism[62℄. Bothofthemarevery

suppressed, espe ially the photon splitting, thus measuring signi ant

magneto-opti al properties of the va uum would be a signal of the existen e of parti les

oupled to two photons. The best onstraints on these phenomena are provided

bythePolarizzazionedelVuoto onLASer(PVLAS)experiment,whoseresultsare

shown in gure 1.4 in the yellow zone labelled PVLAS [63℄. In parti ular, this

refers to the birefringen e measurements, for the di hroism ones are ompletely

superseded by ALPS results.

environments. In the past de ades many white dwarfs with very strong magneti

elds up to

10

10

G were dis overed. We allthese obje ts magnetisedwhite

dwarfs(MWD).Thepolarisationoflight omingfromMWDisstronglyinuen ed

by the magneti eld. A typi alfra tion of 5% of the lightis ir ularlypolarised,

as required to ele tromagneti waves propagating in a magnetised atmosphere,

while the linearly polarised fra tion is something less but of the same order, and

ould be explained by photon-ALP onversion. If ALPs with

m

φ

.

10

−6

eV and

g

φ

&

10

−11

GeV

−1

exist, the linearly polarised fra tion of light would be larger

than 5% for MWDs with

B = 10

9

G, and onsequently this region has to be

ex luded [64℄. The bound an improve by more than an order of magnitude for

the ouplingtophotons upto

g

φ

∼ 10

−12

GeV

−1

ifthedataaboutthe MWD

with the strongest magneti eld,

B

∼ 10

10

G,are onrmed[64℄.

1.4 Where ould axions and ALPs hide?

Untilnowea haxionandALPsear hhas beenunfruitful,providingonlyex lusion

bounds. However, besides the phenomenologi al need to have an axion in order

to solve the strong CP problem, there are some observational hints about whi h

regions of the parameter spa e ould hide an axion or an ALP. We draw dashed

ellipsesingures1.2and1.4tohighlightthe regionsof parameterspa esuggested

by these observations.

Re ently, the measurement of TeV photons from very far sour es  namely

somea tivegala ti nu lei(AGN)haspuzzledastrophysi ists[65,66℄. Veryhigh

energy photons should inelasti ally s atter with the ba kground light produ ing

e

+

_e

−

pairs. They should be rapidly absorbed by the intergala ti medium and

we should observe none of them if oming from very far sour es. Of ourse, it

ould be that the spe trum of the sour e is mu h harder than we expe t or that

we overestimate the amount of ba kground photons in the medium. However, it

ould alsobethat theseTeV photons, intera ting withthe extragala ti magneti

eld, os illate into ALPs, then es ape the absorption and nally, on e onverted

ba k into the photon form, are dete ted by our teles opes [6769℄. To solve this

10

−12



10

−9

, and oupling greater than

g

φ

∼ 10

−12

GeV

−1

[56℄. Of ourse, mu h

about these onsiderations depends onthe assumptions onthe sour e and onthe

extragala ti magneti eld. Espe iallythestrengthofthelatterhastobeassumed

very loseto theupperlimittohaveanappre iableee t. To avoidthisproblem,

one an assume the photon-ALP onversion to happen in the regions around the

sour eand inour galaxy, where the magneti elds are better known [70,71℄. We

have nevertheless to noti e that if the onstraint omingfrom MWD is solid, not

toomu hspa eisleftfor ALPstosolvethe transparen yofthe universeproblem.

Buteven if this is the ase, the ALP-photon onversionae ts the polarisationof

distantastrophysi alsour esand,on esome moreinformationaboutthe oherent

omponents on the intergala ti magneti eld are obtained, it will be possible

to extra t some useful limits from AGNs [72,73℄ and other elestial obje ts, like

quasars [74℄.

Astrophysi s provides anothervery interesting lue related tothe evolution of

white dwarf stars. It seems that if the axion has a dire t oupling to ele trons

and a de ay onstant

f

a

∼ 10

9

GeV, it provides anadditionalenergy-loss hannel

that permits to obtain a ooling rate that better ts the white dwarf luminosity

fun tionthan the standard one [75℄. The sele ted mass rangeis inthe meV range

and

g

aγγ

∼ 10

−12

_GeV

−1

. The hadroni axionwould also help intting the data,

but in this ase a stronger value for

g

aγγ

is required toperturbatively produ e an ele tron ouplingof the requiredstrength.

Finally, axions and ALPs are also perfe t dark matter andidates. We will

devote a large se tionof hapter2 tothis topi .

Afterthe presentationmadeinthis hapter,wewilldeal fromnowonwiththe

osmologi al bounds. Sin e many of the topi s that willbe treated involve both

axions and ALPs, we will refer to them using the term pseudos alars when they

are on the same level. We alsowant to underline that inthe followingdis ussion

Establishing an axion or ALP reli

population

In order to obtain information about the axion and its ALP relatives from

os-mologi al onsiderations it is rst of all ne essary to understand if a primordial

population of pseudos alars an be established. Several me hanisms an a hieve

thistaskandtheywillbedes ribedinse tions2.1and2.2. Se tions2.3and2.4deal

with the ways a pseudos alarpopulation an disappear from the osmi plasma.

Intherest of thedissertationwe willdeal withaat Robertson-Walkermetri

ds

2

= dt

2

_{− R}

2

(t) dr

2

+ r

2

dϑ

2

+ r

2

sin

2

ϑ

2

dϕ

2

,

(2.1)

where

R

is the osmi s ale fa tor, whi h has length dimensions,

t

is the time oordinate,

r

is the dimensionless radial omoving oordinate, and

(ϑ, ϕ)

are the dimensionless omoving angular oordinates. The osmi s ale fa tor is growing

in time, representing the expansion of the universe. A useful way of measuring

the expansion is through the redshift

z

, whi h measures the ratio between the wavelength

λ

e

of a light signal emitted at time

t

e

and the wavelength

λ

d

of the same signal dete ted at time

t

d

, and it isdened tobe

1 + z =

λ

d

λ

e

=

R(t

d

)

R(t

e

)

.

(2.2)

The redshift of asignal measured today is alsoa pra ti alway torefer to

2.35

× 10

−13

GeV, the temperature ata given redshift is easily obtained with the

formula

T = T

0

(1 + z)

if we assumeno heatingof the thermal bath. The expansion rate isdened tobe

H(t)

≡

R(t)

˙

R(t)

(2.3)

where the dot stands for the time derivative. The Friedmann equation links

H

with the energy density of the universe

ρ

,

H

2

=

8π

3

ρ

m

2

Pl

,

(2.4)

the Plan k mass being

m

Pl

= 1.2211

× 10

19

GeV. In the radiation dominated

universe, the energy density is

ρ =

π

2

30

g

∗

(T )T

4

_,

(2.5)

whi h depends onthe temperature

T

, and the equation (2.4) be omes

H

≃ 1.66g

∗

(T )

1/2

T

2

m

Pl

.

(2.6)

We will refer very often to this form for

H

. The

T

dependent quantity

g

∗

is the number of relativisti internal degrees of freedom, whi h is plotted in gure 2.1.

Its denition is

g

∗

(T ) =

X

i=bosons

g

i

 T

i

T



4

+

7

8

X

i=fermions

g

i

 T

i

T



4

,

(2.7)

where the indi es

i

run overthe bosons and fermions with temperature

T

i

and

g

i

internal degrees of freedom, whi h are relativisti when the photon temperature

is

T

.

The riti al energy density

ρ

c

= 3

(H

0

m

Pl

)

2

8π

= 10.5 h

2

keV

cm

3

,

(2.8) where

H

0

= 100 h km s

−1

_Mpc

−1

is the present value of the expansion rate and

h

_{≃ 0.7}

is its present-day normalized value, denes the energy density of a at universe expanding at

H

0

rate. It an be used asaunit of measure for theenergy

10

-6

10

-4

0.01

1

100

1

2

5

10

20

50

100

200

T @GeVD

g

*

HS

L

Figure2.1: Thenumberofrelativisti internal degreesoffreedom

g

∗

(solid)and thenumber of relativisti entropy degrees of freedom

g

∗S

(dashed) as fun tions oftemperature.

density of the dierent onstituents of the universe. Thus, dening the present

ratios

Ω

r

= ρ

r

/ρ

c

,

Ω

m

= ρ

m

/ρ

c

,and

Ω

Λ

= ρ

Λ

/ρ

c

respe tively forradiation,matter andva uumenergy,theexpansionrate anbe onvenientlyexpressedasafun tion

of the redshift with

H(z) = H

0

pΩ

r

(1 + z)

4

+ Ω

m

(1 + z)

3

+ Ω

Λ

,

(2.9)

if there are no rea tions onverting one energy forminto the other.

The entropy density of the radiationdominated universe is

s =

2π

2

45

g

∗S

(T )T

3

_,

(2.10)

and

g

∗S

are the numberof relativisti entropy degrees of freedom,

g

∗S

(T ) =

X

i=bosons

g

i

 T

i

T



3

+

7

8

X

i=fermions

g

i

 T

i

T



3

,

(2.11)

whi hisalsoplottedingure2.1. Foralltheusual osmologi alquantities,like

g

∗

and

g

∗S

,wefollowthedenitionsofTheEarlyUniverse byKolbandTurner[76℄.