Prodution mehanisms II: thermal relis

Welearnedinthe previoussetionthatthehigherthe ALP/axiondeayonstant,

the moreabundant would be the nal yield,it being proportionalto

φ ² _i ∝ f _φ ²

^, ^see

equation (2.22a). If however

f φ

^is ^not ^high ^enough ^to ^guarantee ^a ^suient

non-thermal prodution, the ALP or the axion an be nevertheless a DM omponent

as itwould bemore strongly oupledand more proneto thermalprodution.

A populationof partiles with

g

^internal ^degrees ^of ^freedom ^whih ^ounts

the polarisation and the partile-antipartile states and mass

m

ⁱⁿ ^thermal

equilibrium has numberand energy densities and pressure given respetively by

n eq = g 2π ²

Z ∞ m

√ E ² − m ²

exp (E/T ) ± 1 E dE ,

^(2.38)

ρ eq = g

2π ² Z ∞

m

√ E ² − m ²

exp (E/T ) ± 1 E ² dE ,

^(2.39)

p eq = g

6π ² Z ∞

m

(E ² − m ² ) ^3/2

exp (E/T ) ± 1 dE ,

^(2.40)

if the temperature is

T

^. ^Aording ^to ^the ^fermioni ^or ^bosoni ^nature ^of ^the

partile, the sign

+

^or

−

^holds ⁱⁿ^the ^previous ^formulae. ^In ^the ^ase ^of ^negligible

hemial potential, the previous equations in the relativisti limit

T ≫ m

^for

bosons are

n _eq = ζ(3)

π ² g T ³ ,

^(2.41)

ρ eq = π ²

30 g T ⁴ ,

^(2.42)

p eq = ρ

3 ,

^(2.43)

while forfermions are

n eq = 3 4

ζ (3)

π ² g T ³ ,

^(2.44)

ρ eq = 7 8

π ²

30 g T ⁴ ,

^(2.45)

p _eq = ρ

3 .

^(2.46)

In the non-relativisti limit,

T ≪ m

^, ^the ^quantum ^statistis ^makes ^no ^dierene

and we have

n eq = g m T

2π ³ ₂

exp

− m T

,

^(2.47)

ρ eq = mn ,

^(2.48)

p eq = T n .

^(2.49)

ThedynamisofapartiledistributionisgovernedbytheBoltzmannequation.

TheBoltzmannequationlinksthetotaltimederivativeofadistributionontheleft

hand side to the mirophysis of partile interations that lies on the right hand

side. Theeetsof theuniverse expansionareinluded onthe lefthandside. If

σ _i

is the ross-setion for the sattering of the partile under examination with the

i

^-speies ⁱⁿ ^the ^bath, ^the ^Boltzmann ^equation ^for ^number ^density ^an ^be ^written

˙

n + 3Hn = − X

i

h σ i v i nn ⁱ − n eq n ⁱ _eq

,

^(2.50)

assumingthattheinterationprodutsrapidlythermalise,whihallowsustowrite

theterm

n eq n ⁱ _eq

^on^the^right-hand^side^[119℄. ^The^thermal^average

h σ i v i

^is^obtained

integratinginmomentumthematrixelementfortheinterationproessmultiplied

by the distribution funtions of the partilesinvolved.

Our aim is to alulate the rate at whih an axion or ALP population an

arise from the sattering of SM partiles in the thermal bath during the early

phases of the universe. The axion and ALP self-interation an be negleted,

beingsuppressedby

f _φ ⁻⁴

^. ^Then,^we^an^onsider^only^theinterationswithpartile

whih partilesin the following. Under this assumption,

n ⁱ

^is ^always

n ⁱ _eq

^and ^the

Boltzmann equationsimplies as

˙

n + 3Hn = − Γ (n − n eq ) ,

^(2.51)

where

Γ = P

i Γ i

^is^the ^total ^interation ^rate^and ^the

i

^-th^partial^one ^is^dened ^to

Γ i = h σ i v i n ⁱ _eq

^. ^F^rom ^equation ^(2.51)^we ^infer^that ^if

Γ

^is ^muh^bigger ^than ^the

expansion rate

H

^the ^number ^density ^is ^fored ^to ^follow ^the equilibrium one. If the right hand side beomes negligible,

n

^is ^only ^diluted ^by ^the ^expansion ^of ^the

universe, and hanges through the fator provided by equation (2.21)

n(t) = n(t ∗ )

R(t ∗ ) R(t)

3 .

^(2.52)

We an onsider

Γ/H > 1

^as ^a ^rule ^of ^thumb ^to ^dene ^the ^thermal equilibrium ondition, while if

Γ/H < 1

^the ^speies ^under examination is deoupled. Both

Γ

and

H

^are temperature dependent. When a partilespeies loses thermalontat with the bath beause all its interations freeze out, it is said to deouple, and

this moment is haraterised by the temperature

T fo

^whih ^is ^the ^solution ^of ^the

equation

H(T fo ) = Γ(T fo )

^. ^At ^any ^later ^time ^between ^the ^freezing ^out ^and ^the

deay, the pseudosalar density an be obtained in rst approximation thanks to

equation (2.52)and the dilutionfator (2.21)

n φ (T ) = n ^eq _φ (T fo ) g ∗S (T )T ³

g ∗S (T fo )T _fo ³ .

^(2.53)

Assuming the partile ontent of the standard model, there is a minimum

pseu-dosalar yieldwhihis given by the value of

g ∗ (T _fo > E _EW ) n φ

n γ ≥ 1 2

g ∗S (T )

106.75 ≃ 0.005g ∗ (T ) .

^(2.54)

Oursimpleriterionabout

Γ/H

^is^suient^to^give^a^rstapproximationofthe axion thermal history aording tothe axion sale

f a

^. ^The ^key ^events ^take ^plae

during the radiation domination era, and therefore

H

^an ^be ^expressed ^through

equation (2.6). It is oneptually onvenient to divide the thermal history of the

universe intothreeepohs, lassifyingthem throughthe mainaxioninteration at

• f a & T & Λ QCD

We learnedinthe previoussetion that whenthe universe temperaturefalls

below

f a

^thePQ-symmetryisspontaneouslybrokenandtheaxion,still mass-less,popsup. If

f a

^is^not^too^high,^the^axion^olouredinterationsdepitedin gure2.6 are eient. Theinterationrate is

Γ ∼ (α _s ³ /f _a ² )n col ∝ (α ³ _s /f _a ² )T ³

where

n col

^is ^the ^number ^density ^of ^oloured ^partiles. ^A ^more ^preise

esti-mate of

Γ

^is^given ⁱⁿ ^[120℄. ^The ^ratio

Γ/H ∝ T

^. ^Thus ^at^high

T

^the ^axions

are in equilibrium if the required temperature was ever ahieved and

they deouple when

T

^beomes ^too ^small. ^T^o ^deouple ^during ^this ^phase,

the axionmust have a deay onstant in the range

f _a ≃ 10 ⁸

10 ¹⁰

^Ge^V, ^thus

amass aroundthe 110meV. If this isthe ase, the frationof the universe

energy density in thermalaxions is negligible,

Ω a h ² ∼ 10 ⁻⁶

10 ⁻⁴

^[120℄.

α s a

8πf a

G

q g s q

α s a

8πf a

G

G ^g ^s G

q G

q

a

g s C q q

2f a

Figure 2.6: Relevant axion interations for

T > Λ _QCD

• Λ QCD & T & m π

T ∼ Λ QCD

^gluons^and^quarks^onne. ^At^this ^stage,^if ^not^yet^deoupled,

axionsan stillinterat withpions thanks tothe eetiveLagrangian (1.23)

(gure2.7), with

Γ ∼ T ⁵ /(f π f a ) ²

^[121℄. ^If ^the ^axion ^deay ^onstant ^is

f a ≃ 10 ⁴

10 ⁷

^Ge^V, ^whihⁱⁿ ^terms ^of ^mass ^is

m a ∼ 0.01

100

^e^V, ^axions ^deouple

atthis stage. The axionyieldinthis aseis largerthan inthe previous one,

and inprinipleitan beabundant enoughtoinuene the evolution ofthe

universe. The formation of LSSs like those we observe today requires the

lightpartileswouldpreventthegravitationalbindingofthesmallestlumps.

Together with the CMB measurements, LSS onstrains the axion mass

m a

to be smaller

0.7

^e^V ^and ^thus

f a > 9 × 10 ⁶

^Ge^V ^[121,^122℄. ^In ^gure ^1.2

this upper bound is labelled as Hot DM. This bounds applies to a small

range of axion masses, for its validity eases if the axion is osmologially

unstable. In this ase the osmologialeets of the axionneeds other tools

tobeanalysed, whihwillbedisussed inthe following hapters.

π

π π

a

C _π f _π f _a

Figure 2.7: Axion-pion interation relevant for

Λ _QCD > T > m _π

• T . m π

One the pions beome non-relativisti, their abundane is exponentially

suppressed, as it an be seen from equation (2.47), and the axion-pion

in-teration freezes out. Axions are only left to interat eletromagnetially

with photons and harged partiles. The interation of axions during this

phase is less eient in keeping them in equilibrium, not only beause the

eletromagnetiforeisweakerthan thestrongone, butalsobeausethe

in-teratingspeiesarejustafewandverydilutedbytheexpansion. Figure2.8

shows how a harged partile interats with an axion via the Primako

ef-fet, whih remains ative until the

e ⁺ e ⁻

annihilation at

T ∼ m e

^. ^F^or ^this

interation,

Γ aγγ ∼ αg _aγγ ² n e

^. ^The ^axion ^deay ^is ^also ^an eletromagneti interation, with

Γ = 1/τ

^. ^In ^partiular, ^this ^proess ^is haraterised by a freezingin temperature,sine

τ

^has^no

T

^dependene^and^thus^the^ratio

Γ/H

growswithtime. Axionswithamasslargerthan

O (10)

^ke^V^never^freeze^out,

beause for them

H ∼ 1/τ

^before ^the ^Primako ^interation ^with ^eletrons

deouples [1℄. In this ase, the thermal equilibrium ondition implies their

a ^g ^aγγ γ

γ

e ^q ^e e

Figure 2.8: Relevant axion interation for

T < m _π

Of ourse, if the axion has diret oupling to other SM partiles, like the

eletron as in gure 2.9, the nal yield would be higher, beause, aording to

equation(2.51),a highertotal interation rate

Γ

^means^a ^later^deoupling. ^In^the

ALP ase, we are mainly interested in the interations involving the two-photon

oupling (1.24). Additional interations would delay the deoupling and inrease

the nal abundane.

e γ

e

a

q e C e e

2f a

Figure2.9: Additionalaxion-eletroninteration, whihisrelevantintheDFSZ

modelfor

T < m _π

As already mentioned, the osmologial stability is the seond key fator to

obtain limits on the parameters of pseudosalars from osmologial observables.

If the axion mass is larger than 20 eV, we an see from equation (2.36) that its

lifetimeis shorter than the age of the universe. Considering our shemati

repre-sentation of axionthermal produtionas a funtion of

f _a

^, ^it ^seems ^worthwhile ^to

givesomemoredetailsabout thelastproesses, fortheydeterminethe abundane

of osmologially unstable axions. Moreover, the generalisation to the ALP ase

isstraightforward.

omputed in [123℄ tobe

Γ P = α g aγγ 2

12 T ³

log T ²

m ² _γ

+ 0.8194

,

^(2.55)

where

m γ 2 = 2παT ² /3

^is ^the ^squared ^plasmon ^mass ⁱⁿ ^a relativisti eletron-positronplasmaand

T

^thetemperature. Takingtheratetobeproportionaltothe number density of eletrons,

n e = 3ζ(3)T ³ /π ²

^, ^we ^an ^generalise ^to ^a

multiom-ponent plasma,using instead the eetive number density of harged partiles

n q = X

i

q ⁱ _EM ² n i ≡ ζ(3)

π ² g q (T )T ³ .

^(2.56)

Theparameter

g q (T )

^represents^the^eetive^number^ofrelativistihargeddegrees of freedom. The plasmon mass has also to be orreted by a fator

m γ ∝ g q ^1/2

The Primako interation rate beomes

Γ _P ≃ α g _aγγ ² 12

π ² n _q 3ζ(3)

log

T ² m ² _γ

+ 0.8194

.

^(2.57)

Thefreezing out temperatureforthe Primako interation iseasilyalulated

from

Γ P /H = 1

^, ^whih ^gives

T fo ≃ 11

α g aγγ 2 m Pl

√ g ∗

g q ≃ 123

√ g ∗

g q

10 ⁻⁹ GeV ⁻¹ g aγγ

2 GeV .

^(2.58)

In the ALP ase, for values of

g φ . 2 × 10 ⁻⁹ GeV ⁻¹

^, interations freeze out at temperaturesabovetheeletroweaksale,wherethepartileontentoftheplasma

is somewhat speulative. For instane in the minimal supersymmetri standard

model senario, above the supersymmetry breaking energy sale we have

g ∗ = 228.75

^, ^while ^the ^SM ^alone ^provides ^only ^106.75 relativisti degrees of freedom.

For

g _φ . 10 ⁻¹⁷ GeV ⁻¹

^we^require^a^freeze^out temperatureabovethePlanksale, whihis most likelymeaningless.

CosmologiallystableALPs must not exeed the measured abundaneof DM,

therefore

Ω φ h ² = ρ φ

ρ c

h ² = m φ n φ

ρ c

h ² < Ω DM h ² = 0.11 .

^(2.59)

Using (2.53), we see that ALPs with a mass

m φ = 154

^e^V ^would ^aount ^for ^all

new degrees of freedom (dof) above the EW sale would relax this bound, whih

islinearly sensitive to

g ∗ (T fo )

^, ^by ^a^fator

(106.75 + new dof)/106.75

^. ^This ^means

that

O (100)

^of ^them ^are ^needed ^for ^a^sizeable ^hange.

Im Dokument Cosmological limits on axions and axion-like particles (Seite 55-62)

φ 2 i ∝ f φ 2

f φ

g

m

n eq = g 2π 2

Z ∞ m

√ E 2 − m 2

exp (E/T ) ± 1 E dE ,

ρ eq = g

2π 2 Z ∞

m

√ E 2 − m 2

exp (E/T ) ± 1 E 2 dE ,

p eq = g

6π 2 Z ∞

m

(E 2 − m 2 ) 3/2

exp (E/T ) ± 1 dE ,

T

+

−

T ≫ m

n eq = ζ(3)

π 2 g T 3 ,

ρ eq = π 2

30 g T 4 ,

p eq = ρ

3 ,

n eq = 3 4

ζ (3)

π 2 g T 3 ,

ρ eq = 7 8

π 2

30 g T 4 ,

p eq = ρ

3 .

T ≪ m

n eq = g m T

2π 3 2

exp

− m T

,

ρ eq = mn ,

p eq = T n .

σ i

i

˙

n + 3Hn = − X

i

h σ i v i nn i − n eq n i eq

,

n eq n i eq

h σ i v i

f φ −4

n i

n i eq

˙

n + 3Hn = − Γ (n − n eq ) ,

Γ = P

i Γ i

i

Γ i = h σ i v i n i eq

Γ

H

n

n(t) = n(t ∗ )

R(t ∗ ) R(t)

3

.

Γ/H > 1

Γ/H < 1

Γ

H

T fo

H(T fo ) = Γ(T fo )

n φ (T ) = n eq φ (T fo ) g ∗S (T )T 3

g ∗S (T fo )T fo 3 .

g ∗ (T fo > E EW ) n φ

n γ ≥ 1 2

φ ² _i ∝ f _φ ²

n eq = g 2π ²

√ E ² − m ²

2π ² Z ∞

√ E ² − m ²

exp (E/T ) ± 1 E ² dE ,

6π ² Z ∞

(E ² − m ² ) ^3/2

n _eq = ζ(3)

π ² g T ³ ,

ρ eq = π ²

30 g T ⁴ ,

π ² g T ³ ,

π ²

30 g T ⁴ ,

p _eq = ρ

2π ³ ₂

σ _i

h σ i v i nn ⁱ − n eq n ⁱ _eq

n eq n ⁱ _eq

f _φ ⁻⁴

n ⁱ

n ⁱ _eq

Γ i = h σ i v i n ⁱ _eq

n φ (T ) = n ^eq _φ (T fo ) g ∗S (T )T ³

g ∗S (T fo )T _fo ³ .

g ∗ (T _fo > E _EW ) n φ

Γ ∼ (α _s ³ /f _a ² )n col ∝ (α ³ _s /f _a ² )T ³

f _a ≃ 10 ⁸

10 ¹⁰

Ω a h ² ∼ 10 ⁻⁶

10 ⁻⁴

G ^g ^s G

T > Λ _QCD

Γ ∼ T ⁵ /(f π f a ) ²

f a ≃ 10 ⁴

10 ⁷

f a > 9 × 10 ⁶

C _π f _π f _a

Λ _QCD > T > m _π

e ⁺ e ⁻

Γ aγγ ∼ αg _aγγ ² n e

a ^g ^aγγ γ

e ^q ^e e

T < m _π

T < m _π