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1.4 Where ould axions and ALPs hide?

2.1.2 Before or after ination?

quantum state,

N occupation ∼ (2π) 3 4π/3

n φ,0

m 3 0 δv 3 ∼ 10 42 m 1

m 0

3/2 eV m 0

5/2

,

(2.31)

where we used

R 0 /R 1 ∼ T 1 /T 0 ∼ p

m 1 m Pl /T 0

. If the ALP self-interations are strong enough to ahieve thermalisation and ALP-number-onserving, as argued

in referenes [87,88℄ for the ase of axions, this high oupation number leads

to the formation of a Bose-Einstein ondensate. This ould imprint interesting

signatures inosmologial observations, likegalatiausti rings [8790℄.

The axion realignment prodution is a partiular ase of what is desribed

above. The

Λ

parameter in this ase is

Λ QCD

, and the phase transition that

turns on the axion mass is the QCD phase transition. The axion thermal mass

was rst alulated from the dilute instanton gas approximation [91℄ and more

reently usingthe instanton liquidone [92℄. The theoretial unertainties in these

alulationsmaketheparametersforthethermalaxionmassslightlydierent. For

example,asimpleand reentapproximationinthedilutegas approximation,that

alsoagrees very well athigh temperaturewith the liquid instanton one, is [92℄

m a (T ) = 4.1 × 10 −4 Λ 2 f a

Λ T

3.34

,

(2.32)

for

Λ = 400

MeV. This approximation is valid above

T ∼ 0.1

GeV, sine below

this value the axion mass approahes the zero-temperature value

m a

given by

equation (1.6). The exat shape of the thermal axionmass determines

t 1

, thus it

aets the

m 0 /m 1

ratio inequation (2.22a).

axionenergy density

ρ a

remains

ϕ 2 i

dependent. Assumingaat prioronthe value

of the initial misalignment angle,

ϕ i ∼ O (1)

is the most likely outome. If this

is the ase, it is possible to state that the axion mass should be

m a & 6 µ

eV,

and thus

f a . 10 12

GeV, to avoid DM over-prodution. This is the traditionally quoted lower bound, rst obtained in 1982 [7779℄. It is represented in gure 1.2

with the Cold DM label. Suessive estimates, with a morerened treatment of

the average of the squared initial angle or of the thermal axion mass, give more

severe boundsbut ofthe sameorder. Fortherealignmentontribution, the reent

referene [92℄ gives a limit whih is stronger by one order of magnitude. The

traditional

f a . 10 12

GeV bound is thereforethe most onservative one.

Itseems that this old DM bound ouldbe easilyavoided inthe ase ination

happenedafterthespontaneous symmetrybreaking,ifweaeptane-tunedvery

smallvalue for

ϕ i

. In gure 2.5are plotted the isoontours of

ϕ i

that provide the

whole old DM density, if in equation (2.22b) we take

m 1 = m 0

and

F (T ) = 1

.

However,

ϕ i

has a minimum value due to the quantum utuations generated in

the pseudosalar eld during ination. These unavoidable inhomogeneities in

ϕ i

are of order

δϕ i ∼ H I /(2πf φ )

, where

H I

is the value of the expansion rate at

the endof ination. Thispreludes ne-tuningofthe universe averageof

φ i

below

H I /2π

,andsets aminimumDMabundaneforaspeiedvalueof

H I

. Sinethe

φ

eld is eetivelymassless during inationinthis senario, these inhomogeneities

orrespond to isourvature perturbations of the gravitational potential. This has

been disussed extensively in the literature in the ontext of axions and of string

axions, see forexample [81,93,94℄.

TheWMAP7observationsoftheprimordialdensityutuationsset very

strin-gentonstraintsonisourvatureperturbationsfromwhihoneanobtainanupper

bound on

H I

, assuminga given

f φ

. WMAP measures [86℄

α ≡ h| S 2 |i

h| S 2 |i + h| R 2 |i < 0.077 ,

(2.33)

where

h| S 2 |i ≈ π H 2 φ 2 I 2

i

is the isourvature power spetrum, and

h| R 2 |i

the adiabati

one, whih is generated by the inaton or by other elds. At the pivot sale

k 0 = 0.002 Mpc −1

, WMAP nds

h| R 2 |i = 2.42 × 10 −9

,whih gives the bound

H I . 4 × 10 −5 ϕ i f φ .

(2.34)

j i = 10

0

j i = 10

-2

j i = 10

- 4

j i = 10

-6

j i = 10

- 8

H I = 10 10 GeV H I = 10 9 GeV H I = 10 8 GeV H I = 10 7 GeV H I = 10 6 GeV

Τ<10 17 s

KSVZ Axion

-15 -10 -5 0 5 10

-20 -18 -16 -14

-12 9

11

13

15

17

Log 10 m Φ @eVD

Log 10 g Φ @ GeV - 1 D Log 10 f Φ @ GeV D

Figure 2.5: The anthropi window. The solid lines represent the isoontours

of the values of

ϕ i

that provide the whole old DM density, aording to

equa-tion(2.22b) ,taking

m 1 = m 0

and

F (T ) = 1

. Dashedlinesgivetheisoontoursof

themaximumvalueof

H I

allowed bythelimit (2.34) , alulatedusing thesame

valueof

ϕ i

asbefore. Asa referene, theKSVZaxion parameters areplotted as

a dottedline. In thegrey region ALPsareosmologially unstable.

The lower bound for the reheating temperature is

T RH > 4

MeV [95℄. Linking

naively the two quantities gives

H I ≈ T RH 2 /m Pl > 1.3 × 10 −24

GeV. In the light

of equation (2.34), it is rather a weak bound by itself. However, in the most

ommoninationmodels,

H I

ispreferredto bemuh larger,toallowmehanisms

that require very high initial temperatures of the universe like leptogenesis,

whih needs

T > 10 9

GeV [96℄. The isoontours of the maximum

H I

allowed by

limit(2.34)are plottedingure2.5, wherethe valueof

ϕ i

that providesthe whole

DMdensityisused. Agivenvalueof

H I

exludestheregionoftheparameterspae

that lies on the right of its isoontour. Then, along the

H I

isoontour

f φ

selets

the

ϕ i

that is provided by primordialutuations,while onthe left half-planethe

initialmisalignmentallowed is largerand thusless ne-tuned.

In the ase of the axion, the values of

f a

that satisfy the onstraint (2.34) for

large

H I

are above the GUT sale, whih seems very reasonable from the model

building point of view: they formthe so-alled anthrophi window 1

. Considering

thattheaxionwasintroduedtoavoidne-tuningtosolvethestrong-CPproblem,

the legitimay of the hoie of a very small

ϕ i

to justify an axion sale linked to

Λ GUT

orthe stringsaleisquestionable[97℄. However, theanthropipointof view an nd a vindiation regarding the DM density and other signiant physial

quantities simply as observational data, from whih it is possible to extrat the

value ofthe initialmisalignmentwithanaposteriorianalysis[98℄. This proedure

islegitimatebeauseoftheintrinsirandomnessof

ϕ i

,andonvertsthene-tuning

issue into anobservationalproblem.

Ontheotherhand,iftheSSBtookplaeafterinationthe

φ i

pikeduprandom

values indierentausally disonneted regions of the universe andthe anthropi

windowloses itssigniane. If thisisthe ase, alsothe gradienteets shouldbe

taken intoaount. Atlarger sales, the DM density averages to aonstant value

orresponding to

h φ 2 i i ∼ π 2 f φ 2 /3

, bearing the mentioned

O (1)

orretion due to

the non-harmonibehaviouroflarge initialphases. Theinitialsize ofthe domains

of dierent

φ i

annotbe larger than

L i,dom ∼ 1

H SSB ∼ m Pl f φ 2 p

g ∗ (f φ ) .

(2.35)

Non-linear eets, due to the attrative self-interation aused by higher order

terms inthe expansion of the potential (1.25), drive the overabundanes to form

peuliar DM lumps that are alled minilusters [99102℄. These at like seeds

enhaningthe suessivegravitationallumpingthatleads tostruture formation.

The miniluster mass is set by the dark matter mass inside the Hubble horizon

d H = H −1

when the self-interation freezes-out, i.e.

M mc ∼ ρ φ (T λ )d H (T λ ) 3

for

1

Theadjetiveanthropirefersto the fat that theexistene ofthe humanspeies requires

theuniverseto fullsomeonditionsaboutitsageandomposition,whih tooabundantaxion

the freeze-outtemperature

T λ

. Long-range interations willbeexponentially sup-pressed at distanes longerthan

1/m φ

so we an expet

T λ

to be of the order

T 1

,

withatmostalogarithmidependeneonotherparameters. Thisisindeedthease

forQCDaxions,forwhihtheminilusteringisquenhedsoonaftertheQCDphase

transition that turns on the potential (1.25) [87℄ giving

M mc ∼ 10 −12 M ⊙

, where

M ⊙ = 1.989 × 10 33 g

is the solar mass, and a radius

R mc ∼ 10 11

m [103℄. In the

aseofALPs,

M mc

anbelargerifthemassislighter. Theauthorsof[102℄pointed

outthatthepresentdataontheCDMpowerspetrumonstrain

M mc . 4 × 10 3 M ⊙

whihtranslates intoalowerbound intemperature

T λ > 2 × 10 −5 GeV

andinthe

ALP mass

m φ > H(T = 2 × 10 −5 GeV) ∼ 10 −20

eV. If some of these minilusters survive the tidal disruption duringstruture formationthey shouldbeobservable

inforthominglensing experiments [102,103℄.