ALP bounds

4.1 Eetive number of neutrinos

4.1.1 ALP bounds

At the beginning of this hapter we have desribed some limiting ases to

un-derstand the matters related to dilution and photon injetion. The key quantity

was

T

^. ^But ^onsidering ^the ^large ^ALP ^parameter ^spae ^and ^all ^the intermediate ases it inludes, it is unavoidable to numerially solve the problem. The ative

speiestoonsiderare photons,eletronsand neutrinos. Eletrons atsas

messen-gers between photons and neutrinos. Photonsthermalise veryfast with eletrons,

whih,throughweakreations,heattheneutrinostokeeptrakofthetemperature

hangesinthe eletromagnetibath. Themostrelevantoftheseenergy

redistribu-tion proesses is

e ⁺ e ⁻ → νν ¯

^. ^We ^neglet^sattering ^proesses ^suh ^as

e ^± ν → e ^± ν

sinethey annothangetheneutrinonumberand arelesseetiveintransferring

energy to the neutrino bath 1

. The energy ow from ALPs to neutrinos an then

be modelled by a set of Boltzmann equationsfor omoving energies, dened as

X i = ρ i R ⁴ ,

^(4.10)

Assume that eletronshave temperaturelarger than neutrinos like in our ase. Then the

speedoftheenergytransferperunitvolumeinsatteringproessesisproportionalto

h δE i T e ⁴ T ν ⁴

with

h δE i ∼ (T e − T ν )

^athermal-averagedenergytransferpersattering. Forannihilationsitis

∝ T ^e (T ν ⁸ − T e ⁸ )

^,^whih^is^muh ^more^sensitive^tononequilibrium situations.

andtheALPphasespaedistributionfuntion

f (K φ )

^as^a^funtion^of^the^omoving

momentum,

K φ = Rk φ

d

d t f (K _φ ) = − (c _γ + c _P )(f − f ^eq ) ,

^(4.11a)

d

d t (X γ + X e ) = 3Hδp e R ⁴ +

Z d ³ K φ

(2π) ³ W φ (c γ + c P )(f − f ^eq ) + Γ eν

R ⁴

h c e X _ν ² _e − X _ν ^{eq 2} _e + c µτ

X _ν ² _µτ − X _ν ^{eq 2} _µτ i

,

^(4.11b)

d

d t X ν e = − Γ eν

R ⁴ c e X _ν ² _e − X _ν ^{eq 2} _e

,

^(4.11)

d

d t X ν µτ = − Γ eν

R ⁴ c µτ

X _ν ² _µτ − X _ν ^{eq 2} _µτ

,

^(4.11d)

d

d t R = 1 R

s 8π

3m ² _Pl X γ + X e + X ν e + X ν µτ + ρ φ R ⁴

,

^(4.11e)

where

ω φ = q

k φ 2 + m φ 2

is the ALP energy and

W φ = ω φ R

^. ^The ^set ^of

equa-tions (4.11) desribes the evolution of the omoving energy density stored in

γ

togetherwith

e ^±

^, ^the^three^speies ^of

ν

^'s,^the^ALPs^and ^the^osmi^sale ^fator

R

Thesolutionofequation(4.11a)isthe distributionfuntion

f

^for^thepseudosalar partile. The ollisionterms for the deay and Primako proesses are [1℄

c γ = m ² _φ − 4m ² _γ m ² _φ

m φ

ω φ

1 + 2T k φ

log 1 − e ^−(ω ^φ ^+k ^φ ^)/2T 1 − e ^−(ω ^φ ^−k ^φ ^)/2T

1 τ ,

^(4.12)

c P ≃ α g _φ ²

16 n e log

"

1 + [4ω φ (m e + 3T )] ² m ² _γ [m ² _e + (m e + 3T ) ² ]

#

.

^(4.13)

Sinetheenergystoredineletronsandpositronsistransferreddiretlytothe

pho-tonbath,wekept

X γ

^and

X e

^togetherⁱⁿ^equation^(4.11b). ^The^term

δp e = p e − ρ e /3

aounts of the omoving energy density gain as eletrons beome inreasingly

non-relativisti,experiening the transitionfrom

ρ e ∝ R ⁻⁴

^to

ρ e ∝ R ⁻³

^. ^Here,

p e

and

ρ e

^are ^the ^pressure ^and ^energy ^density ^of

e ^±

^. ^The ^seond ^term ⁱⁿ ^the ^right

hand side rekons up the photon-ALPenergy exhange, while in the lastlast line

of equation (4.11b) are the neutrino-eletron interations. The equations (4.11)

and(4.11) provide the evolutionof neutrino omovingenergies. Neutrinoavour

inuenetheeletron-neutrinoenergyexhange rate. Wehaveseparatedthe three

urrent interations with eletrons. The rst is the ase of

X ν e

^for ^eletron

neu-trinos, while in the seondone we have

X ν µτ

^for ^muon ^and ^tau ^avours ^together.

The energy exhange rate is given by the fator

Γ eν ≡ G ² _F T γ

^, ^with

G F

^the ^F^ermi

onstant, multiplied by

c e ≃ 0.68

^for ^the ^eletron ^avour ^and

c µτ ≃ 0.15

^for ^the

muon and tau ones, whih follow from the appropriate thermally averaged ross

setion. Weassumethatneutrinosalwayshaveathermaldistribution,determined

onlybyaneetivetemperature,whihshouldbeareasonablerstapproximation

and aurate enough for our purposes. We negletthe energy reshuing between

dierent neutrino speies, whih does not inuene the total neutrino density at

leading order. Finally, through equation (4.11e) we alulate the expansion rate

ofthe universethrough theevolutionoftheosmisalefator

R

^. ^The^initial

on-ditions are speiedat

T ≫ MeV

^by^having ^all^speies ^at ^a^ommontemperature and the ALPnumberdensity given by equation (2.53).

Forvalues

g φ . 10 ⁻⁷ GeV ⁻¹

^,^the^Primako^proess^is^deoupledⁱⁿ^the

temper-aturerangeof interestandanbenegleted. Iftheinverse deay isalsonegligible,

weanintegratethe ALPphasespaedistributionexpliitlyanddiretlyompute

the evolutionof the number density

d

d t n φ R ³

= − n φ R ³

τ

^(4.14)

instead of using equation (4.11a). In this way we reover the exponential deay

law

N _φ ∝ e ^−t/τ

^. ^The ^integralⁱⁿ ^equation ^(4.11b)^is ^then

Z d ³ K φ

(2π) ³ W φ (c γ + c P )(f − f ^eq ) ≃ m φ n φ R ⁴ /τ .

^(4.15)

WehavesannedtheALPparameterspaein[2℄,andwepresentourresultsfor

N eff

ⁱⁿ^the

g φ

τ

^and

m φ

τ

^planes ⁱⁿ^gure^4.3. ^W^e^required ^a^standard ^osmology

senarioattemperaturesbelowthestandardmatter-radiationequality

T mr ∼ 1

^e^V.

Thuswe have analysedonlythe part ofparameter spaewhere

T d > T mr

^,ⁱⁿ^order

tonot perturbthestandardpiturewiththedeay. Fromequation2.63,weobtain

a mass lower bound for the validity of our alulations

g φ

GeV ⁻¹ & 10 ⁻³ eV

m _φ 3/2

,

^(4.16)

0.01 0.1

0.5 0.5

1 1.5 2.11

2.3 2.4 2.5

N eff 3

- 5 -3 -1 1 3 5

- 3 - 1 1 3 5 7 9

Log ₁₀ m _Φ @MeVD Log 10 Τ @ s D

(a)

0.01

0.1

0.5 0.5 1

1.5

2.11

2.3 2.4 2.5 3

N eff

-14 -12 -10 -8 -6 - 4

- 3 - 1 1 3 5 7 9

Log ₁₀ g Φ @GeV ^-1 D Log 10 Τ @ s D

(b)

Figure4.3: Contourplotsoftheeetivenumber ofneutrinos

N _eff

^as^a^funtion

oftheALPmassandlifetime(left)andoftheouplingparameter

g _φ

^and^lifetime

τ

^(right). ^ALP ^osmologies^leading ^to

N _eff < 2.11

^an ^be ^safely^exluded.

whih largely overlaps with the limits obtained from CMB spetral distortions

obtained inthe previous hapter.

The sharp features of the isoontours in gure 4.3are due to the abrupt and

sizeable derease of

g ∗S

^during ^the ^QCD ^phase transition. The reli abundane of ALPsdepends on

g φ

^through ^the ^freezing ^out temperature, and for ALPs with oupling around

g _φ ∼ 10 ^−7.5 GeV ⁻¹

^we ^have

T _fo ≃ Λ _QCD

^from ^equation ^(2.58).

In a small range of

g _φ

^the ^ALP ^abundane onsiderably hanges, aording if the deoupling is late enough for ALPs to share the entropy transferred fromthe

QCDdegreesoffreedomornot. ThelargeristheALPabundane,thelargeristhe

subsequent entropy release at the time of the deay, hene the peuliar shape of

theisoontours. Ingure4.3a weobservethesame features,asthe

g φ

^dependene

ishidden in

τ

In gure 4.4, we present some illustrativeexamples depiting the evolution of

the

X

^s ^of ^eletrons, ^neutrinos ^and ^ALPs ^as ^a ^funtion ^of ^the temperature. Note that in all of them, when ALPs beome non-relativisti, the ratio

X _φ /X _γ

^rises

beause it beomes proportional to

m φ /T

^, ^until ^the ^age ^of ^the ^universe ^beomes

omparable with

τ

Figure 4.4a shows a typial ase of very early deay of a massive ALP, when

neutrinos are still partially oupled to eletrons. This example depits the ALP

behaviourinthe lowerrightornerofgure4.3a,andorrespondinglyinthelower

leftoneofgure4.3b. Here

N eff = 2.6

^,^marginally^dierent^from^the^standard^value

3

^. ^Even ^if^the ^ALP^energy ^dominates^the ^universe ^and^the ^entropy ^injeted

du-ring the deay ishuge, the reheating temperatureis large enoughfor neutrinos to

almost fully reovertheir thermalabundane. In general,the nal value of

N eff

^is

related not to the total entropy released, but only to the part of it injeted after

the freeze out of the neutrino-eletron interations. In this partiular region of

the parameter spae,the neutrino dilutionismainlysensitiveto theALP lifetime

and not to

m φ

^or

g φ

individually. The outome of a deay earlierthan

t ∼ 10 ⁻²

is undistinguishable from standard osmology, sine neutrinos regain ompletely

their thermal abundane. Around

T ∼ m e

^, ^eletrons ^and ^positrons ^beome

non-relativisti and annihilate, heating the photon bath but not the neutrinos, whih

have deoupled. The ratio

X ν /X γ

^therefore ^dereases. ^In ^this ^period,^the ^photon

temperature inreases with respet to the neutrino temperature by the standard

fator

(4/11) ^1/3

^due ^to^entropy onservation.

The piture hanges onsidering later deays, after neutrino deoupling:

pho-tons andeletrons get allthe ALP enthropy,asshown ingure4.4b. Thisiswhat

happensingure4.3a andingure4.3b abovethe thik isoontour. The neutrino

dilution is omputable in this ase, as the ratio of the nal and initialomoving

entropies of the photon-eletron bath is given by equation (4.7). The

tempera-ture of the eletromagneti bath inreases with respet to the neutrino one by a

fator

(S f /S i ) ^1/3

^, ^making

N eff = 3(S f /S i ) ^−4/3

^beause ^also ^the ^eletron ^entropy

ends up in photons. The neutrino energy density is therefore strongly diluted by

the energy gain of photons plus eletrons. Note that this is mainly a funtion of

m φ √

τ ∝ g φ √ m φ

−1

as we an evine from equation (4.7) whih produes

the harateristi slope of the isoontours at long ALP lifetimes

τ ∝ m φ −2

gure 4.3a and

τ ∝ g _φ ⁴

ⁱⁿ ^gure ^4.3b. ^F^or

g φ . 10 ⁻⁹ GeV ⁻¹

^, ^this ^is ^the ^only

dependene on

g φ

^, ^sine

Y φ

^is ^onstant. ^Another ^example ^of ^late ^ALP ^deay, ^but

with a smaller mass, is shown in gure 4.4. In this ase we observe rst the

e ^±

m _Φ = 81 GeV g _Φ =H10 ¹³ GeVL ^-1 Τ= 25 ms N _eff = 2.60

10 ^-1 1 10 10 ²

T @MeVD

X _i X _Γ 2

(a)

m _Φ =9.4 GeV g _Φ =H10 ¹² GeVL ^-1 Τ= 0.15 s N eff = 0.45

10 ^-1 1 10 10 ²

T @MeVD

X _i X _Γ 2

(b)

m _Φ = 32 keV g _Φ =H6.3´10 ⁸ GeVL ^- ¹ Τ= 1.6´10 ⁹ s N eff = 0.26

10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 1 10 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

T @MeVD

X i

X _Γ 2

()

m _Φ = 32 keV g _Φ =H4´10 ⁵ GeVL ^-1 Τ= 631 s

N eff = 2.34

10 0 ^-3 10 ^-2 10 ^-1 1 10

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

T @MeVD

X i

X _Γ 2

(d)

Figure 4.4: Examples of the evolution of the omoving energy of all speies

of neutrinos (solid), ALPs (dashed) and eletrons (double-dashed), as funtions

of the temperature of the universe. All energies are normalized to one thermal

annihilation, whih heats photons with respet to neutrinos. A sizeable neutrino

dilutionis observable afterthe ALP deay.

Finally,ingure4.4dweshowanexampleforwhihinversedeaysarerelevant,

whih is on the left side of gure 4.3a and on the right in gure 4.3b. As the

temperature drops, we observe a rst derease of the ALP energy density due to

the eletrons heating the photon bath. The inverse deay hannel opens around

T ∼ 70

^keV ^and ^helps ^ALPs ^to ^regain equilibrium before disappearing at

T ∼ m φ

^. ^During rethermalization, the photon energy dereases, whih an be seen

as a slight rise in

X ν /X γ

^. ^Due ^to ^this ^mehanism, ⁱⁿ ^the ^small-mass ^and

short-lifetimeregionoftheparameterspaewehaveseenthatentropyonservationgives

N eff = 3(11/13) ^4/3 ≃ 2.4

^. ^If

m φ

^is^larger^than^a^few^Me^V,^the deay-in-equilibrium happenswhenneutrinosarestilloupled,so

N eff

^approahes^3. ^Thedisappearane from the thermal bath is governed only by the mass, and the isoontours of

N eff

exatly followthe isoontoursof

m φ

The exluded region of the parameter spae are determined omparing these

numbers with the limits (4.9), taking the 99% C.L. value of

N eff > 2.11

^. ^W^e ^are

ompelledtothis onservativehoie beause the 95%C.L. valueof

N eff > 2.39

^is

justbelowthe

N eff ≃ 2.4

^obtained^from^the^ase^ofdeay-in-equilibrium. Given all the unertainties of this alulation, we an onsider this large part of parameter

spaedisfavouredbut notexluded. Ingure4.3, theexludedregionisabovethe

thik line, whih is oloured in yellow in the summarising plot for this hapter,

gure 4.10.

Im Dokument Cosmological limits on axions and axion-like particles (Seite 89-95)

4.1 Eetive number of neutrinos

4.1.1 ALP bounds

T

e + e − → νν ¯

e ± ν → e ± ν

X i = ρ i R 4 ,

h δE i T e 4 T ν 4

h δE i ∼ (T e − T ν )

∝ T e (T ν 8 − T e 8 )

f (K φ )

K φ = Rk φ

d

d t f (K φ ) = − (c γ + c P )(f − f eq ) ,

d

d t (X γ + X e ) = 3Hδp e R 4 +

Z d 3 K φ

(2π) 3 W φ (c γ + c P )(f − f eq ) + Γ eν

R 4

h c e X ν 2 e − X ν eq 2 e + c µτ

X ν 2 µτ − X ν eq 2 µτ i

,

d

d t X ν e = − Γ eν

R 4 c e X ν 2 e − X ν eq 2 e

,

d

d t X ν µτ = − Γ eν

R 4 c µτ

X ν 2 µτ − X ν eq 2 µτ

,

d

d t R = 1 R

s 8π

3m 2 Pl X γ + X e + X ν e + X ν µτ + ρ φ R 4

,

ω φ = q

k φ 2 + m φ 2

W φ = ω φ R

γ

e ±

ν

R

f

c γ = m 2 φ − 4m 2 γ m 2 φ

m φ

ω φ

1 + 2T k φ

log 1 − e −(ω φ +k φ )/2T 1 − e −(ω φ −k φ )/2T

1

τ ,

c P ≃ α g φ 2

16 n e log

"

1 + [4ω φ (m e + 3T )] 2 m 2 γ [m 2 e + (m e + 3T ) 2 ]

#

.

X γ

X e

δp e = p e − ρ e /3

ρ e ∝ R −4

ρ e ∝ R −3

p e

ρ e

e ±

X ν e

X ν µτ

Γ eν ≡ G 2 F T γ

G F

c e ≃ 0.68

c µτ ≃ 0.15

R

T ≫ MeV

g φ . 10 −7 GeV −1

d

d t n φ R 3

= − n φ R 3

τ

N φ ∝ e −t/τ

Z d 3 K φ

e ⁺ e ⁻ → νν ¯

e ^± ν → e ^± ν

X i = ρ i R ⁴ ,

h δE i T e ⁴ T ν ⁴

∝ T ^e (T ν ⁸ − T e ⁸ )

d t f (K _φ ) = − (c _γ + c _P )(f − f ^eq ) ,

d t (X γ + X e ) = 3Hδp e R ⁴ +

Z d ³ K φ

(2π) ³ W φ (c γ + c P )(f − f ^eq ) + Γ eν

R ⁴

h c e X _ν ² _e − X _ν ^{eq 2} _e + c µτ

X _ν ² _µτ − X _ν ^{eq 2} _µτ i

R ⁴ c e X _ν ² _e − X _ν ^{eq 2} _e

R ⁴ c µτ

X _ν ² _µτ − X _ν ^{eq 2} _µτ

3m ² _Pl X γ + X e + X ν e + X ν µτ + ρ φ R ⁴

e ^±

c γ = m ² _φ − 4m ² _γ m ² _φ

log 1 − e ^−(ω ^φ ^+k ^φ ^)/2T 1 − e ^−(ω ^φ ^−k ^φ ^)/2T

c P ≃ α g _φ ²

1 + [4ω φ (m e + 3T )] ² m ² _γ [m ² _e + (m e + 3T ) ² ]

ρ e ∝ R ⁻⁴

ρ e ∝ R ⁻³

e ^±

Γ eν ≡ G ² _F T γ

g φ . 10 ⁻⁷ GeV ⁻¹

d t n φ R ³

= − n φ R ³

N _φ ∝ e ^−t/τ

Z d ³ K φ

(2π) ³ W φ (c γ + c P )(f − f ^eq ) ≃ m φ n φ R ⁴ /τ .

GeV ⁻¹ & 10 ⁻³ eV

m _φ 3/2

Log ₁₀ m _Φ @MeVD Log 10 Τ @ s D

Log ₁₀ g Φ @GeV ^-1 D Log 10 Τ @ s D

N _eff

g _φ

N _eff < 2.11

g _φ ∼ 10 ^−7.5 GeV ⁻¹

T _fo ≃ Λ _QCD

g _φ

X _φ /X _γ

t ∼ 10 ⁻²

(4/11) ^1/3

(S f /S i ) ^1/3

N eff = 3(S f /S i ) ^−4/3

τ ∝ g _φ ⁴

g φ . 10 ⁻⁹ GeV ⁻¹

e ^±