Before or after ination?

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

^Me^V. ^This approximation is valid above

T ∼ 0.1

^Ge^V, ^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 ⁱⁿ^equation ^(2.22a).

axionenergy density

ρ a

^remains

ϕ ² _i

^dependent. ^Assuming^a^at ^prior^on^the ^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

^e^V,

and thus

f a . 10 ¹²

^Ge^V, ^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

^Ge^V ^bound ^is ^therefore^the ^most onservative one.

Itseems that this old DM bound ouldbe easilyavoided inthe ase ination

happenedafterthespontaneous symmetrybreaking,ifweaeptane-tunedvery

smallvalue for

ϕ i

^. ^In ^gure ^2.5^are ^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

^,^and^sets ^a^minimum^DM^abundane^for^a^speied^value^of

H I

^. ^Sine^the

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

^, ^assuming^a ^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

^. ^Dashed^lines^give^the^isoontours^of

themaximumvalueof

H I

^allowed ^by^the^limit ^{(2.34) ,} ^alulated^using ^the^same

valueof

ϕ _i

^as^before. ^As^a ^referene, ^the^KSVZ^axion ^parameters ^are^plotted ^as

a dottedline. In thegrey region ALPsareosmologially unstable.

The lower bound for the reheating temperature is

T RH > 4

^Me^V ^[95℄. ^Linking

naively the two quantities gives

H I ≈ T _RH ² /m Pl > 1.3 × 10 ⁻²⁴

^Ge^V. ^In ^the ^light

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

ommoninationmodels,

H I

^is^preferred^to ^be^muh ^larger,^to^allow^mehanisms

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

whih needs

T > 10 ⁹

^Ge^V ^[96℄. ^The ^isoontours ^of ^the ^maximum

H I

^allowed ^by

limit(2.34)are plottedingure2.5, wherethe valueof

ϕ i

^that ^provides^the ^whole

DMdensityisused. Agivenvalueof

H _I

^exludes^the^region^of^the^parameter^spae

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

H I

^isoontour

f φ

^selets

the

ϕ i

^that ^is ^provided ^by ^primordial^utuations,^while ^on^the ^left ^half-plane^the

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

^or^the ^string^sale^isquestionable[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

^,^and^onverts^the^ne-tuning

issue into anobservationalproblem.

Ontheotherhand,iftheSSBtookplaeafterinationthe

φ i

^piked^up^random

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

^annot^be ^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

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

^an^be^larger^if^the^mass^is^lighter. ^The^authors^of^[102℄^pointed

outthatthepresentdataontheCDMpowerspetrumonstrain

M mc . 4 × 10 ³ M ⊙

whihtranslates intoalowerbound intemperature

T λ > 2 × 10 ⁻⁵ GeV

^andⁱⁿ^the

ALP mass

m _φ > H(T = 2 × 10 ⁻⁵ GeV) ∼ 10 ⁻²⁰

^e^V. ^If ^some ^of ^these minilusters survive the tidal disruption duringstruture formationthey shouldbeobservable

inforthominglensing experiments [102,103℄.

Im Dokument Cosmological limits on axions and axion-like particles (Seite 48-52)

1.4 Where ould axions and ALPs hide?

2.1.2 Before or after ination?

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

,

R 0 /R 1 ∼ T 1 /T 0 ∼ p

m 1 m Pl /T 0

Λ

Λ QCD

m a (T ) = 4.1 × 10 −4 Λ 2 f a

Λ T

3.34

,

Λ = 400

T ∼ 0.1

m a

t 1

m 0 /m 1

ρ a

ϕ 2 i

ϕ i ∼ O (1)

m a & 6 µ

f a . 10 12

f a . 10 12

ϕ i

ϕ i

m 1 = m 0

F (T ) = 1

ϕ i

ϕ i

δϕ i ∼ H I /(2πf φ )

H I

φ i

H I /2π

H I

φ

H I

f φ

α ≡ h| S 2 |i

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

h| S 2 |i ≈ π H 2 φ 2 I 2

i

h| R 2 |i

k 0 = 0.002 Mpc −1

h| R 2 |i = 2.42 × 10 −9

H I . 4 × 10 −5 ϕ i f φ .

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

ϕ i

m 1 = m 0

F (T ) = 1

H I

ϕ i

T RH > 4

H I ≈ T RH 2 /m Pl > 1.3 × 10 −24

N occupation ∼ (2π) ³ 4π/3

m ³ ₀ δv ³ ∼ 10 ⁴² m 1

R ₀ /R ₁ ∼ T ₁ /T ₀ ∼ p

m ₁ m _Pl /T ₀

m a (T ) = 4.1 × 10 ⁻⁴ Λ ² f a

ϕ ² _i

f a . 10 ¹²

f _a . 10 ¹²

α ≡ h| S ² |i

h| S ² |i + h| R ² |i < 0.077 ,

h| S ² |i ≈ _π ^H ² _φ ² ^I ²

h| R ² |i

k 0 = 0.002 Mpc ⁻¹

h| R ² |i = 2.42 × 10 ⁻⁹

H _I . 4 × 10 ⁻⁵ ϕ _i f _φ .

j ⁱ = 10

j ⁱ = 10

j ⁱ = 10

j ⁱ = 10

j ⁱ = 10

Τ<10 ¹⁷ s

Log ₁₀ m _Φ @eVD

ϕ _i

m ₁ = m ₀

ϕ _i

H I ≈ T _RH ² /m Pl > 1.3 × 10 ⁻²⁴

T > 10 ⁹

H _I

h φ ² _i i ∼ π ² f _φ ² /3

H SSB ∼ m _Pl f _φ ² p

d H = H ⁻¹

M mc ∼ ρ φ (T λ )d H (T λ ) ³

M mc ∼ 10 ⁻¹² M ⊙

M ⊙ = 1.989 × 10 ³³ g

R mc ∼ 10 ¹¹

M mc . 4 × 10 ³ M ⊙

T λ > 2 × 10 ⁻⁵ GeV

m _φ > H(T = 2 × 10 ⁻⁵ GeV) ∼ 10 ⁻²⁰