Realignment mehanism

In the realignment mehanism a eld, whih in the early universe an take a

random initial state, rolls down towards the minimum of the potential. One it

has reahed the bottom,itovershootsthe minimumand startstoosillatearound

it. Ifthe quantaof the eld are osmologiallystable, theseosillationsbehaveas

a old darkmatter uid. Their energy density infat is dilutedby the expansion

of the universe as

ρ ∝ R ⁻³

^.

The simplest exampleis that of a salar eld

φ

^{of mass}

m φ

^{with Lagrangian}

L = 1

2 ∂ µ φ∂ ^µ φ − 1

2 m ² _φ φ ² + L I ,

^(2.12)

where

L ^I

^enodes interationsofthesalareldwith itselfandtherestofpartiles inthe primordialbath. In general, the mass reeivesthermal orretions from

L ^I

whih might be ruial,thus

m φ = m φ (t)

^{should be} understood. In eah ausally onnetedpathofthe universe thesalar eldhas aninitialvalue,

φ _i

^{. Ifination}

0 0.5 1 5 10 -0.5

0.0 0.5 1.0

Log ₁₀ Htt ₁ L Log ₁₀ Hm _Φ HL

ΦΦ _i

Figure 2.2: Numerial solution of equation (2.13) (solid) and theevolution of

log ₁₀ (m _φ /H)

^{(dashed) for the valuesused inthe equation.}

alreadyhappened,

φ

^{isuniformlyequalto}

φ i

^{inthewholeobservableuniverse. The}

equation of motion in the expanding universe for the homogeneousomponent of

φ

^{alled the zero mode isobtained negleting the gradient eets,}

φ ¨ + 3H φ ˙ + m ² _φ φ = 0 .

^(2.13)

Its solutionanbeseparatedintotworegimes. Inarst epoh,

3H ≫ m _φ

^,so

φ

^is

anoverdamped osillatorand getsfrozen,

φ ˙ = 0

^{. Atalatertime,}

t 1

^,haraterized by

3H(t 1 ) = m φ (t 1 ) ≡ m 1

^{, the damping beomes} underritial and the eld an roll down the potential and starts to osillate. During this epoh, the mass term

is the leading sale in the equation and the solution an be found in the WKB

approximation,

φ ≃ φ i

m 1 R ³ ₁ m φ R ³

1/2

cos Z t

t 1

m φ dt

,

^(2.14)

where

φ(t 1 ) ∼ φ i

^{, sine up to}

t 1

^{the eld evolution is frozen. Notethat toobtain}

this solution only the denition of

H = ˙ R/R

^{has been taken into aount, not}

its atual time dependene, and so it is valid for radiation, matter, and vauum

energy dominatedphases ofthe universe and theirtransitions. Figure2.2shows a

The approximate solution (2.14) orresponds to fast osillations with a slow

amplitude deay. Dening this amplitude

A (t) = φ i (m 1 R ³ ₁ /m φ R ³ ) ^1/2

^{and the}

phase

α(t) = R t

m φ (t)dt

^{, the energy density of the salar eld is}

ρ φ = 1

2 φ ˙ ² + 1

2 m ² _φ φ ² = 1

2 m ² _φ A ² + ... ,

^(2.15)

where the dots stand for terms involving derivatives of

A

^{, whih by assumption}

are muh smallerthan

m φ

^{, beause}

m φ ≫ H

^{in this regime. The pressure isthen}

p φ = 1

2 φ ˙ ² − 1

2 m ² _φ φ ² = − 1

2 m ² _φ A ² cos (2α) − A A ˙ m φ sin (2α) + ˙ A ² cos ² (α) .

^(2.16)

When the eld just starts to osillate the equation of state is a non-trivial and

strongly time dependent funtion. However, at muh later times,

t ≫ t ₁

^{, the}

osillationsin the pressure our at time sales

1/m _φ

^{, muh faster than the}

os-mologialevolution. Wean thereforetakeanaverage overthese osillations,and

the pressure is then

h p φ i = h A ˙ ² cos ² (α) i = 1

2 A ˙ ² .

^(2.17)

At leading orderin

A ˙ /( A m)

^{, the equation of state is just}

w = h p i / h ρ i ≃ 0 ,

^(2.18)

whih is exatlythat of non-relativisti matter.

It follows fromequation (2.14) that the energy density ina omovingvolume,

ρ φ R ³

^{,is not onserved if the salar mass hanges intime. However, the quantity}

N _φ = ρ φ R ³

m φ

= 1

2 m ₁ R ₁ ³ φ ² _i ,

^(2.19)

isonstant,andanbeinterpretedasaomovingnumberofnon-relativistiquanta

ofmass

m φ

^{. Weanuse this} onservationtoompute the energydensitytodayto obtain

ρ φ (t 0 ) = m 0

N φ

R ³ ₀ ≃ 1

2 m 0 m 1 φ ² _i R 1

R 0

3

,

^(2.20)

wherequantities with a

0

^{-subsript are evaluated atpresent time.}

Through the onservation of omovingentropy

S

^{, the dilution fator}

(R 1 /R) ³

inequation (2.20) an be expressed as

R 1

R 3

= g ∗S (T )T ³

g ∗S (T 1 )T ₁ ³ .

^(2.21)

10 ^-5 0.001 0.1 10 1000 0.2

0.4 0.6 0.8 1.0

T @GeVD

F H T L

Figure 2.3: Thefuntion

F (T )

^{,usedinequations (2.22) .}

Using the expression for the Hubble onstant in the radiation dominated era,

equation (2.6), and the denition of

T 1

^,

3H(T 1 ) = m 1

^,

T 1

^{an bewritten interms}

of

m 1

^{andthePlankmass. Inthisway,equation(2.20)forthematterdensityfrom}

the misalignementmehanismtodayan thenbeexpressed inamore quantitative

way as

ρ φ,0 ≃

0.17 keV cm ³

r m 0

eV r m 0

m 1

φ i

10 ¹¹ GeV 2

F (T 1 ) ,

^(2.22a)

Ω φ ≃ 0.016 h ⁻²

r m 0

eV r m 0

m ₁

φ i

10 ¹¹ GeV 2

F (T 1 ) ,

^(2.22b)

where

F (T 1 ) ≡ (g ∗ (T 1 )/3.36) ^{3 4} (g ∗S (T 1 )/3.91) ⁻¹

^{isa smooth funtion ranging from}

1

^to

∼ 0.3

^{in the interval}

T 1 ∈ (T 0 , 200GeV)

^{, and it is plotted in gure 2.3. The}

quantity

Ω φ = ρ φ,0 /ρ c

^{is the fration of energy density of the universe in ALPs.}

The abundane ismost sensitiveto the initialamplitude of the osillations, being

proportionalto

φ ² _i

^{,andtoalesserdegreetothepresent massvalue}

m 0

^{. Thefator}

1/ √ m 1

^{reetsthedampingofthe}osillationsintheexpandinguniverse: thelater the osillationsstart,i.e.thesmaller

T 1

^{and therefore}

H 1

^and

m 1

^{,the lessdamped}

they are for agiven

m ₀

^.

and other large sale struture probes[86℄,

ρ CDM = 1.17 keV

cm ³ , Ω CDM = 0.11 h ⁻² ,

^(2.23)

it seems that only very large values of

φ i

^{an provide a}

ρ φ,0

^{that aounts for all}

the dark matter. However, a relativelysmall

φ i

^{ouldbe ompensated by a small}

m ₁ ≪ m ₀

^.

Itisnowneessary toprovideamodelforthethermalmassofthepseudosalar

partiletoompare the estimate(2.22a)with theobserved abundane (2.23). For

thepotential(1.25),theALP satisestheequationofmotion(2.13)aslong asthe

angle

ϕ ≡ φ/f φ

^{is small. The inauray of the quadrati} approximation an be anyway ured by an additional orretion fator to (2.22a). This is normally an

O (1)

^{fator exept if we ne tune the initialondition to}

ϕ = π

^.

Figure 2.4represents the region of the parameter spae of ALPs in whih the

realignamentmehanismouldprovideasuientprodutionofDM.Theallowed

regionsof ALPdarkmatterinthe

m φ

g φ

^{planeingure2.4anbeobtained using}

φ _i = ϕ _i αC γ

2πg φ

(2.24)

with

ϕ i = φ i /f φ

^,theinitialmisalignmentangle,whoserangeisrestritedtovalues between

− π

^and

+π

^{. The modeldependent fator}

C γ

^{willfrom now on be taken}

to be unity for presentation purposes, but the reader should keep in mind that

in priniple it an assume a very dierent value aording to the model it omes

from. To tune the rightDM abundanewe have used the a priori unknown value

of

ϕ i

^{. The upper bound on}

g φ

^{reets the fat that}

ϕ i

^{annot be larger than}

π

^{, and thus assumes}

ϕ i ∼ π

^{. Moving to lower values of}

g φ

^{requires ination}

happening after SSB in order to have a homogeneous small value of

ϕ i

^{whih is}

inreasingly ne-tuned to zero to avoid over-abundant DM prodution. In this

sense the values losest to the boundary, orresponding to the largest values of

the photon oupling, an be onsidered the most natural ones. We will further

ommenton the ne-tuning issue later on.

In the simplest realization of an ALP model, the mass reeives no thermal

orretions, thus

m φ

^{is onstant throughout the universe expansion and the DM}

yieldan be inferred diretly from equation (2.22a) using

m 1 = m 0

^{. In gure 2.4}

itis the dark red region labelled

m 0 = m 1

^.

HB excluded

Haloscope excluded

Τ<10 ¹⁷ s

m ₁ >3HHT _eq L

m ₁ =m ₀ m ₁ m ₀ =HLTL ^Β

Β= 0 Β= 1 Β= 3 Β= 5 Β= 7 Β= 9

Axion models

-8 -6 -4 -2 0 2 4 6

-16 -14 -12 -10 -8 -6

Log ₁₀ m _Φ @eVD Log 10 g Φ @ GeV - 1 D

Figure 2.4: Axion and ALP DM from therealignment mehanism. The axion

modelband is plotted asin gure1.4. The regions ofthe ALP parameter spae

in whih the realignment mehanism provides suient or too large DM

abundaneareplotted indierentredgradationsfordierentthermalmass

mod-els. The pink region fulls theminimumrequirement for non-thermal produed

ALP to behave like CDM. The greyregion over the part ofparameter spaein

whihtheALPlifetime, whihisprovidedbyequation(2.36) ,isshorterthanthe

age of the universe. The turquoise regions have been tested by halosopes and

no ALPs have been found. As a referene, theHB upperbound is plotted as a

If the global symmetry assoiated to the ALP is anomalous, then the mass is

providedby theinstantonipotentialjustasthe

η ^′

^{orthe axionaquiretheirmass}

thankstoQCDinstantons. TheALP needsinprinipleanotherunbroken

SU(N )

group,whihondenses atasale

Λ

^{. If}

T

^isthetemperatureofthe

SU(N )

^setor,

then the thermalALP mass is

m φ ≃



 

 

 

 

Λ ^′2

f φ ≡ m 0

^for

T ≪ Λ , m 0 Λ ^′′

T

β

for

T ≫ Λ .

(2.25)

It makes sense to naively assume

Λ ^′ ∼ Λ ^′′ ∼ Λ

^{, while the preise relation among}

these sales depends on the details of the model. At temperatures larger than

Λ

^, eletri-sreening damps long range orrelations in the plasma and thus the instantoni ongurations, resulting in a derease of the ALP mass. In spei

models the exponent

β

^{an be obtained for instane from instanton} alulations, but here it is left as a free parameter. Assuming the onset of ALP oherent

osillations to happen in the mass suppression regime, it is easy to obtain an

expression for

m 0 /m 1

^{whih is the expeted enhanement inthe DM abundane,}

r m 0

m 1

=

√ m 0 m Pl

Λ ^′′

_β+2 ^β

(3 × 1.66 √

g ∗1 ) ^{2β+4 −β}

^(2.26)

and the fator that ontrols the enhanement is

√ m 0 m Pl

Λ ^′′ ∼ Λ ^′ Λ ^′′

r m Pl

f φ

.

^(2.27)

These models an provideonly a moderate enhanement of the DM density with

respettotheonstant

m φ

^{ase. ThegainedregionsfortheALPDMaseforvalues}

of

β = 1, 3, 5, 7, 9

^{an be seen in gure 2.4 from bottom to top the lowermost}

region

m 1 = m 0

orresponds, of ourse, to

β = 0

^{. Atually, even onsidering}

unrealistially huge values of

β

^{does not help muh, as an be seen from the}

asymptotiapproahof the highest

β

^{ases. This isreeted by the nite limitof}

equation (2.26) when

β → ∞

^{, but it follows from its denition, equation (2.25).}

In the

β → ∞

^limit,

m φ

^{is a step funtion of} temperature,

m φ ∝ Θ(T − Λ)

^{, and}

the relation

m ₀ = Λ ² /f _φ

^determines

Λ

^from

m ₀

^and

f _φ

^{. Thus eah point in the}

m φ

g φ

^{parameter spae has an impliit maximum DM abundane,} independent of

β

^{. The ruial assumption that leads to these onlusions is that}

Λ ^′ ∼ Λ

^,

beauseitdoesnot allowtoonsider arbitrarysmallvaluesfor

Λ

^{foragiven mass.}

Therefore, models in whih

Λ ^′ ≫ Λ

^{imply generially higher DM abundane and}

require smaller initial amplitudes

φ i

^. Unfortunately, at the moment we an not provide afully motivated example.

Finally,theminimalrequirementforanALPtobehaveasCDMisthatatlatest

at matter-radiation equality, at a temperature

T _eq ∼ 1.3

^{eV, the mass attains its}

urrent value

m 0

^{and therefore the DM density starts to sale truly as}

1/R ³

^{. In}

partiular, at this point the eld should already have started to osillate. This

orresponds toalowerlimiton

m 1

^,

m 1 > 3H(T eq ) = 1.8 × 10 ⁻²⁷

^{eV,whihimplies}

an upper bound on

ρ φ,0

^,

ρ φ,0 < 1.17 keV cm ³

m 0

eV

φ i

54 TeV 2

.

^(2.28)

In other words, if we want these partilesto be the DM, we need

(m ₀ /eV)(φ _i /54 TeV) ² > 1 ,

^(2.29)

givingaonstraintontherequiredinitialeldvalueasafuntionofthemasstoday,

whih is plotted in gure 2.4 in pink. Again, we are not able to provide a well

motivatedmodel for anALP that ould perform this late realignmentprodution

and this limithas to be onsidered just asthe broadest region of ALP parameter

spae that in prinipleould provide the right amountof CDM.

Thedarkmattergeneratedbytherealignmentmehanismhasinteresting

prop-erties beyond thoseof olddarkmatter. Atthe timeof theirprodution, partiles

from the realignmentmehanism are semi-relativisti. Their momenta are of the

order of the Hubble onstant

p ∼ H 1 ≪ T 1

^{. Aordingly we have today}

out-side of gravitational wells a veloity distribution with a very narrow width of

roughly,

δv(t) ∼ H 1

m ₁ R 1

R ₀

≪ 1 .

^(2.30)

Combined with the high number density of partiles,

n φ,0 = N φ /R ³ ₀ = ρ CDM /m 0

^,

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).}

Im Dokument Cosmological limits on axions and axion-like particles (Seite 40-48)