Pseudosalar deay

The freezingin of the deay makes the pseudosalarsregain thermalontat with

the bath. The deay temperature

T _d

^an ^be^obtained ^from

τ H = 1

^, ^so^we ^need^to

know the energy density of the universe inorder toalulate

H

^. ^If pseudosalars do not dominate the universe energy budget when they deay,

T d

^is

T d ≃ 0.6 g ∗ ^1/4 (T d )

g φ

10 ⁻⁷ GeV ⁻¹

m φ

MeV 3/2

MeV .

^(2.63)

If insteadthe pseudosalarenergy density does dominate, whih happens when

φ

isverynon-relativistiand faroutofequilibrium,thedeaytemperatureisinstead

T d ≃ 0.7

[g ∗ (T _d )/g ∗ (T _fo )] ^1/3

g φ

10 ⁻⁷ GeV ⁻¹

4/3 m φ

MeV 5/3

MeV .

^(2.64)

This is typiallylarger than the previousase, sine a matterdominated universe

expands moreslowlythan aradiationdominatedone. Sinethe universe beomes

radiationdominated afterthe deay, the temperaturein equation(2.63) gives the

orret orderof magnitude for the reheating temperature.

If the deay temperature

T _d

^is ^smaller ^than

3m _φ

^, ^the pseudosalar reahes thermal ontat with the bath being non-relativisti. From equation (2.47) and

(2.53), we know that

n eq (T d ) ≪ n φ (T d )

^, ^and ^thus ^the ^Boltzmann ^equation ^(2.51)

beomes

˙

n + 3Hn = − n

τ .

^(2.65)

Thesolutioniseasilyfoundtobe

n(t) = n(t d ) (R d /R) ³ exp ( − (t − t d )/τ )

^,^where

t d

and

R d

^are respetively the time and the sale fator when

T = T d

^. ^The ^number

density rapidly dereases beause of the deay, and two photons per deaying

pseudosalar are reated.

On the other side, if

T _d & 3m _φ

n _eq (T _d )

^an ^not ^be ^negleted, ^and ^the ^ALP

an regain the thermal abundane in this ase it would be given by equation

(2.41),and sohigherthan

n φ (T d )

^thanks^to^the ^inverse ^deay ^proess,

γγ → φ

The rate for this proess is

Γ γγ→φ ≃ 1 τ

m ² _φ − 4m ² _γ m ² _φ

D m φ

ω

E ,

^(2.66)

where

h m φ /ω i

^is ^the ^thermally^averaged ^time ^dilatation^fator, ^being

ω ≥ m φ

^the

energy of the outoming pseudosalar. Deay or inverse deay is only

kinemati-ally allowed if

m _φ < 2m _γ

^. ^F^or

T ≫ m _e

^, ^we ^have

m _γ ∼ T

^while ^for

T ≪ m _e

e ⁺ e -annih.

m ^a = 3T

G ΓΓ®a >H G _P >H

G _C >H

10 ^-1 1 10 10 ² 10 ³ 10 ⁴

10 ^-1 1 10 10 ²

10 ^-1 1 10 10 ² 10 ³ 10 ⁴

10 ^-1 1 10 10 ²

T @keVD m a @ keV D

Figure 2.11: Axion deoupling and reoupling (

C _γ = 1.9

^, ⁱⁿ ^the ^light ^yellow

region it is also

C _e = 1/6

^). ^Thik ^solid ^line: ^freeze-out ^of ^Primako ^proess.

Medium solid line: oupling and freeze-out of theCompton proess. Thin solid

line:reouplingofinversedeay. Intheyellowshadedregion,axionsareinthermal

equilibrium. The light yellow region is relevant only if the Compton proess is

eetive. The dashed line denotes

m _a = 3T

^, ^on ^the ^left ^of ^it ^axions ^are

non-relativisti. The vertial linesdelimitthe

e ⁺ e ⁻

annihilationepoh.

we have

m γ ≪ T

^, ^thus ^the deay/inverse deay hannels open up not far from

m φ ∼ max { T, m e }

^. ^In ^this ^ase, ^some ^photons ^are ^subtrated ^from ^the

ther-mal bath to make the pseudosalars regain the equilibrium distribution. The

pseudosalar population then follows the equilibrium distribution, and beomes

Boltzmann suppressed, basially disappearing from the bath, when the

tempera-turedropsbelow

m _φ

^. ^Thisequilibriumdeay ofrelativistipseudosalars,together withthe previousaseofnon-equilibriumdeay ofnon-relativistipartiles,should

bewell kept in mindas they willbeakey point tounderstandthe topis of

hap-ter 4.

Fousing onthe axionase, adiretouplingwith theeletron an berelevant

ornotaordingtothemodel. Sine

Γ _P

τ

^and^eventually^the^Compton^interation

411 10 ^-1 1 10 10 ²

0 0.5 1

m _a @keVD n a n a eq T = 0.1 m e

Figure2.12: Axionnumberdensity

n _a

^after

e ⁺ e ⁻

annihilationfromnumerially solving the Boltzmann equation until

T = m _e /10

^. ^The equilibrium density

n ^eq _a

isdened intermsofthephotontemperature. Solidline: onlyPrimakoproess

(

C _γ = 1.9

^). ^Dashed ^line: ^Primako^and ^Compton⁽

C _γ = 1.9

C _e = 1/6

^).

rate

Γ C ∼ αC _e ² n e /f a 2

are unambiguously determined as a funtion of

f a

^or

m a

one a model is hosen, it is possible to plot the freezing out and freezing in

temperatures, like in gure 2.11, where the parameters hosen are

C γ = 1.9

^and

C _e = 1/6

^. ^The ^yellow ^area ^represents ^the ^range ⁱⁿ ^whih ^the ^axion interations are eient, while below the dashed line the axion is relativisti. Therefore, the

inverse deay is ative in the yellow area delimited by the dashed line and the

thin solid one, along whih

Γ γγ→a = H

^. ^F^rom ^this ^gure ^it ^is ^lear ^that ^the

lower is

f a

^, ^the ^higher

m a

^, ^and ^thus ^the ^later ^the ^axion ^deouples, ⁱⁿ ^partiular

above

10

20

^keV ^this ^never ^happens. ^In ^gure ^2.12 ^we ^plot ^the ^ratio ^between

the axion number density and its equilibrium value, both at

T = m e /10

^. ^It ^is

obtained solving numerially the set of Boltzmann equations for axions, photons

and eletrons. Inluding the Compton interation, the resulting yield is higher,

for the axion, having an additional interation hannel, deouples later so it its

abundane is less diluted by the fator (2.21). The range mostly aeted by the

further oupling is

m a = 0.5

¹⁰ ^e^V. ^We ^an ^see ⁱⁿ ^gure ^2.11 ^that ^is ^where ^lies

the Compton light yellow region, and from this plot we an better appreiate

and 2.12 inhapter4, where the Boltzmannequations willbe disussed.

Wehaveshowninthis hapterhowapopulationofpseudosalarsanariseand

then disappear in the earlyuniverse. Now itis time todevelop itsonsequenes.

Signals from the sky: reli deay photons

Eah pseudosalar deay produes a ouple of photons, as we have seen in the

previoushapters. Thefateof thesephotonsdependsmainlyatwhihstageofthe

universe evolution the deay takesplae.

The many harged partiles in the primordial plasma eiently satter the

photonspropagatingamongthem. Theuniverseisthereforeoptiallyopaqueuntil

itbeomesoldenoughtopermitthe eletronswhihsurvived the

e ^±

annihilation toombinewithnuleitoformneutralatoms. Thisevent,knownasreombination,

happens rather late in the history of the universe. The temperature has to ool

down toa value aroundaoupleof ordersof magnitude belowthe bindingenergy

of hydrogen and helium atoms. This delay trend is very peuliar of ombination

events in osmology, as it aets also primordial nuleosynthesis. The reason is

the overwhelming numberof photons ompared withthe numberof baryons. The

baryon-to-photon ratio ismeasured tobe [86℄

η = n b

n γ

= (6.23 ± 0.17) × 10 ⁻¹⁰ .

^(3.1)

The number of photons whose energy is above the photo-dissoiation threshold

therefore anbeeasilylargerthan

n b

^. ^The ^ombination^event ^has ^to^wait^behind,

untilthishighenergytailofthephotondistributionemptiesbeauseoftheooling.

The reombination is a ruial event for our understanding of the history of

the universe. Just after this epoh the osmi mirowave bakground (CMB) is

released. Mostoftheinformationwehaveabouttheearlyuniverseisextrapolated

In this hapter we address the eets of the deay photons injeted after the

universe beomes transparent or immediately before. In the rst ase they an

freelypropagate andinpriniplebedeteted, while inthe otherthey shouldleave

animprintonthe CMB. We willanalyse alsothe eet of the pseudosalar deay

onthe ionizationhistory of the universe.

3.1 Spetral distortions of the osmi mirowave

bakground

ThephotondistributionismaintainedinthermalequilibriumbeauseofCompton

sattering, double Compton sattering and bremsstrahlung interations with the

residual eletrons. This means that if a perturbing event happens, the photon

distributionan evolve againto the equilibriumondition,but only ifthese

inter-ations have enough time to operate. Compton sattering

γ + e ⁻ ↔ γ + e ⁻

^has

the fastest rate among these proesses,

Γ ∝ α ²

^, ^but ^it ^an ^only ^lead ^to ^kineti

equilibrium. Sine it onserves the photon number, it an not erase a hemial

potentialinthedistribution. Double Comptonsattering

γ + e ⁻ ↔ γ + γ + e ⁻

^and

bremsstrahlung

e ⁻ + p ⁺ ↔ e ⁻ + p ⁺ + γ

^are omparatively slower,

Γ ∝ α ³

^. ^They

hange the number of photons, and thus they permit to ahieve the full thermal

equilibriumondition, i.e.a Plankian spetrum [128℄.

FIRAS measured the CMB spetrum inthe range 221 m

−1

, and found itto

beis aperfet blak-body withinverysmallexperimentalerrors, onlyan

O (10 ⁻⁵ )

deviation is allowed by data [129℄. Thus, any perturbation of the photon

equi-librium ondition must have been smaller than the measurement unertainties,

or the perturbing event must our long enough before the bremsstrahlung or

double-Comptonease to be eetive.

Beause of the very small value of

η

^, ^the bremsstrahlung proess is subdom-inant with respet to double Compton until

e ⁺ e ⁻

annihilation. The exat value of the freezing out temperature for eah photon-eletron interation depends on

the wavelength. Double Compton sattering is eetive at all wavelengths until

the temperature drops below

T DC ∼ 750

^e^V ^at

t DC

^. ^At temperatures lower than

T _DC

^double ^Compton ^and bremsstrahlung an still produe or absorb photons,

but only if their frequeny is

ω ≪ T

^. ^Compton ^sattering redistributes pho-tons along the whole spetrum and deouples later, at

T C ∼ 25

^e^V. ^Therefore, ^if

a photon injetion is not ompletely reabsorbed before

t DC

^, ^the ^resulting ^CMB

spetrum follows a Bose-Einstein distribution at high frequeny and a Plankian

one in the frequeny range where double Compton and bremsstrahlung are still

ative [128℄. The residual degeneray parameter is onstrained by FIRAS to be

| µ | < 0.9 × 10 ⁻⁴

^[129℄.

After

t DC

^, ^the ^photons ^follow ^a ^random ^walk ^path ^sattered ^by ^the ^eletrons.

Sine the average photon energy is very smallompared with the eletron mass,

the eletron reoil is negligible and photons simply boune o in random

dire-tions. Photons emitted at this stage distort the overall photon spetrum in a

very peuliar way. Eletrons rapidly thermalise with the non-thermal population

of photons and their eetive temperature inreases. However, as we said, CMB

photonsannotgainenergyeientlyoutofthem. TheCMBspetrumishowever

slightly inuened by the heated eletrons. They dissipate the exeeding thermal

energy pushing few photons towards higher frequeny and this imprints a typial

patternonthe CMBspetrum. Thisphenomenon isdesribedby theKompaneets

equation [130,131℄

∂n γ

∂y = 1 x ²

∂

∂x

x ⁴ ∂n γ

∂x + n γ + n γ 2

.

^(3.2)

Thevariable

x = ω/T e

^is^the^ratio^between^the^photon^frequeny

ω

^and^the^eletron

temperature

T e

^. ^The ^parameter

y

^is^given ^by

dy = T e (t) − T (t)

m e

σ T n e (t)dt ,

^(3.3)

where

n e

^is^the ^eletron ^number ^density,

T

^the ^photon temperature and

σ T

^is ^the

Thomson ross setion. A stationarysolution of equation (3.2)is

n(x) = 1

e ^x+µ − 1 ,

^(3.4)

beause of the number onserving Compton interation. If

T e ≫ T

^, ^the

Kompa-neets equation simplies[130℄,

∂n _γ

∂y = 1 x ²

∂

∂x x ⁴ ∂n _γ

∂x

^(3.5)

andiftheperturbationtotheequilibriumisnottoobigtheapproximatedsolution

gives

δn γ

n γ

= y x e ^x e ^x − 1

x e ^x + 1 e ^x − 1 − 4

,

^(3.6)

whose limiting ase are

− 2y

^for

x ≪ 1

^and

yx ²

^for

x ≫ 1

^[130,^131℄. ^Computing

the exat spetrum requires to alulate the eet of the deay on

T e

^and

nume-rially integrate equation (3.2). Like in the

µ

^ase, ^the observational onstraint is very limiting,

| y | < 1.5 × 10 ⁻⁵

^[129℄. ^Of ^ourse, ^all ^of ^this ^depends ^on ^the

rate of photon-eletroninterations. Attemperaturearound

T _dec = 0.26

^e^V, ^most

eletrons and protons are ombined into neutral hydrogen, the universe beomes

eetively transparent and the CMB deouples.

Thephotoninjetionduetopseudosalardeayisanexampleofaperturbation

stritly onstrainedby FIRASdata. The non-thermalontribution tothe photon

spetrumfrom deay isa peak entred atenergy

ω ∼ m φ /2

^. ^Assuming ^a^thermal

population, fromequation (2.54) we have

ρ φ

ρ _γ & 0.007 m φ

T ,

^(3.7)

wherewehavetaken

g ∗S = 3.9

^. ^The^energy^injetion^is^always^greater^than

10 ⁻⁵ ρ γ

sine the ratio

m φ /T > 1

^, ^and ^thus ^it ^is potentially in radial onit with ob-servations. To onventionallyesape this limitonlya small frationof the energy

injetion should inuene the photon distribution, thus the deay has to be just

started when CMB is released or to our long before reombination. However,

we will see in the followingthat there are other osmologial data sets whih

ex-ludepseudosalars able tolear the CMB distortionhurdle through late orearly

enoughdeay. ThesefurtherlimitslargelyoverlapwiththeCMBdistortionbound.

Therefore, here we onsider suient only a rough estimate of the pseudosalar

parameter spae whih is exluded by CMB spetral distortions. A osmologial

senariowith

g ∗ (T fo ) & O (10 ³ )

^would ^permit^to^evade unonventionallythisCMB onstraint. Also the dilution of the pseudosalar density due to some onsistent

entropy injetion duringthe epohinwhih

φ

^is^deoupled ^from^the ^thermal^bath

an in priniple relax this bound. However, this has to happen muh before

pri-mordial nuleosynthesis, otherwise it an be onstrained by the same arguments

95% C.L.

2 4 6 8 10 12 14 16 18 20

10 ^-6 10 ^-5 10 ^-4 10 ^-3 10 ^-2

m a @keVD

È Μ H ¥ LÈ

-7 - 6

-5 -4 -3 -2 -¥

-5 -4

-3

-2

6 8 10 12 14 16 18

10 ^-1 1

m a @keVD

∆

Figure 3.1: Left: Photon

µ

^parameter ^after ^axion ^deay ⁽

δ = 1

^). ^The

obser-vational upper bound is indiated. Right: The ontours of

log ₁₀ | µ |

ⁱⁿ ^the

δ

m _a

plane where blak/red orresponds to positive/negative values of

µ

^. ^The ^thik

blakline orrespondsto the boundary

µ = 0

^. ^The^shaded^area ^is^exluded.

Ingure3.3whihgivesasummaryofthishapterthelightgreenregions

labelled CMB

µ

^and^CMB

y

^orresponds ^to

T DC > T d > T C

^and

T C > T d > T dec

respetively. The inuene of later deays requires a dierent treatment and itis

disussedinthefollowingsetions. Wealwaysassumedthattheamountofphotons

injetedisalwaystoolargetobethermalised,sinewearenotonsideringherethe

unonventional senario of pseudosalar dilution just disussed. Note that these

bounds are somewhatonservative. The deay isnot aninstantaneousevent,soa

signiant amount of energy an be released after

T d

^. ^This ^is ^espeially ^true ^for

m φ &

^ke^V,^for ⁱⁿ^this ^ase ^thepseudosalarenergy density dominatesthe universe beforethe deay andthe subsequent perturbationofthe photonspetrumishuge.

In the axion ase, we have numeriallysolved the dierential equation for the

evolution of the degenerayparameter

µ

^of ^the ^photondistribution[1℄. Assuming

µ

^small, ^its ^evolution ^is ^governed ^by ^the ^equation^[128℄,

dµ

dt = dµ a

dt − µ (Γ DC + Γ B ) ,

^(3.8)

where

Γ DC

^and

Γ B

^are^the^double^Compton^andbremsstrahlungratesfordereasing

µ

^, ^whih ^are ^the ^inverse ^of ^the ^relaxation ^times ^for ^the ^two ^proesses. ^The

rate-of-hange due tothe axiondeay photoninjetion isgiven by

d µ a

dt = − 2 2.14

3 ρ _γ

d ρ a

dt − 4 n _γ

dn a

dt

,

^(3.9)

where

d n a /dt = − n a /τ

^and

d ρ a /dt = m a d n a /dt

^. ^Our ^results ^are ^shown ⁱⁿ

-gure 3.1, where we plot the nal

µ

^value ^as ^a ^funtion ^of

m a

^and

δ ≡ C γ /1.9

^for

hadroni axions. In the left panel we xed

C γ = 1.9

^, ^the ^value ⁱⁿ ^the ^simplest

KSVZ model, and in this ase we found

m _a > 8.7 keV

^at ^95% ^C.L.

.

^(3.10)

This is a robust bound, sine

µ

^is ^a ^steep ^funtion ^of

m a

^. ^The ^CMB ^distortion

eetdependssensitivelyontheaxion-photoninterationstrength for

δ < 1

^(right

panel of gure 3.1). Generally the spetral distortions get larger for smaller

C γ

at a given

m a

^, ^beause ^if ^the ^deay ^happens ^later ^the ^photon distribution is less protetedagainstdistortions. Forlarge

C γ

^, ^the ^nal

µ

^hanges ^sign^from^negative

to positive with inreasing

m _a

^. ^F^or

δ < 0.1

µ

^is ^always ^positive ^sine ^axions

deaynon-relativistially,thustheenergyinjetionismoreimportantthanphoton

number, see equation (3.9). Ofourse, beause of the sign hange in

µ

^some

ne-tunedases existwhere the nal

µ

^an ^be ^aiden^tally ^zero.

Im Dokument Cosmological limits on axions and axion-like particles (Seite 64-74)

T d

τ H = 1

H

T d

T d ≃ 0.6 g ∗ 1/4 (T d )

g φ

10 −7 GeV −1

m φ

MeV 3/2

MeV .

φ

T d ≃ 0.7

[g ∗ (T d )/g ∗ (T fo )] 1/3

g φ

10 −7 GeV −1

4/3 m φ

MeV 5/3

MeV .

T d

3m φ

n eq (T d ) ≪ n φ (T d )

˙

n + 3Hn = − n

τ .

n(t) = n(t d ) (R d /R) 3 exp ( − (t − t d )/τ )

t d

R d

T = T d

T d & 3m φ

n eq (T d )

n φ (T d )

γγ → φ

Γ γγ→φ ≃ 1 τ

m 2 φ − 4m 2 γ m 2 φ

D m φ

ω

E ,

h m φ /ω i

ω ≥ m φ

m φ < 2m γ

T ≫ m e

m γ ∼ T

T ≪ m e

e + e -annih.

m a = 3T

G ΓΓ®a >H G P >H

G C >H

10 -1 1 10 10 2 10 3 10 4

10 -1 1 10 10 2

10 -1 1 10 10 2 10 3 10 4

10 -1 1 10 10 2

T @keVD m a @ keV D

C γ = 1.9

C e = 1/6

m a = 3T

e + e −

m γ ≪ T

m φ ∼ max { T, m e }

m φ

Γ P

τ

411

10 -1 1 10 10 2

0 0.5 1

m a @keVD n a  n a eq T = 0.1 m e

n a

e + e −

T = m e /10

n eq a

C γ = 1.9

C γ = 1.9

C e = 1/6

Γ C ∼ αC e 2 n e /f a 2

f a

m a

C γ = 1.9

C e = 1/6

Γ γγ→a = H

f a

T _d

T d ≃ 0.6 g ∗ ^1/4 (T d )

10 ⁻⁷ GeV ⁻¹

[g ∗ (T _d )/g ∗ (T _fo )] ^1/3

10 ⁻⁷ GeV ⁻¹

T _d

3m _φ

n(t) = n(t d ) (R d /R) ³ exp ( − (t − t d )/τ )

T _d & 3m _φ

n _eq (T _d )

m ² _φ − 4m ² _γ m ² _φ

m _φ < 2m _γ

T ≫ m _e

m _γ ∼ T

T ≪ m _e

e ⁺ e -annih.

m ^a = 3T

G ΓΓ®a >H G _P >H

G _C >H

10 ^-1 1 10 10 ² 10 ³ 10 ⁴

10 ^-1 1 10 10 ²

10 ^-1 1 10 10 ² 10 ³ 10 ⁴

10 ^-1 1 10 10 ²

C _γ = 1.9

C _e = 1/6

m _a = 3T

e ⁺ e ⁻

m _φ

Γ _P

411

10 ^-1 1 10 10 ²

m _a @keVD n a n a eq T = 0.1 m e

n _a

e ⁺ e ⁻

T = m _e /10

n ^eq _a

C _γ = 1.9

C _γ = 1.9

C _e = 1/6

Γ C ∼ αC _e ² n e /f a 2

C _e = 1/6

e ^±

= (6.23 ± 0.17) × 10 ⁻¹⁰ .

γ + e ⁻ ↔ γ + e ⁻

Γ ∝ α ²

γ + e ⁻ ↔ γ + γ + e ⁻

e ⁻ + p ⁺ ↔ e ⁻ + p ⁺ + γ

Γ ∝ α ³

O (10 ⁻⁵ )

e ⁺ e ⁻

T _DC

| µ | < 0.9 × 10 ⁻⁴

∂y = 1 x ²

x ⁴ ∂n γ

e ^x+µ − 1 ,

∂n _γ

∂y = 1 x ²

∂x x ⁴ ∂n _γ

= y x e ^x e ^x − 1

x e ^x + 1 e ^x − 1 − 4

yx ²

| y | < 1.5 × 10 ⁻⁵

T _dec = 0.26