Diret detetion of reli deay photons

and madetheonservativeassumption thateahphotonemittedduringthedeay

an ionize only one atom. On the left panel the mass is xed,

m φ = 100

^eV, ^and

the lifetimevaries. Of ourse,for longer lifetimes the ALP ionising eet appears

later. Onthe rightpanelthe lifetimeisinsteadxed to

10 ²²

^s,^and^the^mass ^varies

from 50 eV to 1 keV. The higher the mass, and onsequently the energy

ω

^of ^the

emitted photons, the less eient is the ionising eet. The one-eletron atom

photoionisationross setion issuppressed for very high energy photons [136℄,

σ ph

ion ∼ 256π 3

α em

Z ²

E 1s (Z) ω

7/2

a ² ₀ ,

^(3.14)

where

Z

^is ^the ^atomi ^number,

E 1s = 13.6Z ²

^eV ^the ^energy ^of ^the

1s

^state,

a 0 = (αm e ) ⁻¹ = 5.292 × 10 ⁻⁹

^m^is ^the ^Bohr ^radius ^and

ω

^the ^photon ^energy^.

TosantheALPparameterspae,weomputed theoptialdepthinthe

inter-val

z = 6

100

^, ^requiring ^it ^to^not ^exeed

τ 6

^. ^We ^made ^two ^dierent alulations, assuming the ALP thermal abundane in [2℄, and seondly that ALPs onstitute

the wholeDM in[3℄. Our resultsare exludingthe lightgreen regionlabelled

x _ion

in gure 3.3, where the thermal origin of ALPs is onsidered. A similar result

was obtained in [3℄. This bound would inrease up to one order of magnitude at

the largest masses for whih ionization is eetive,

m φ . 300

^eV, ^if ^we ^assume

optimistially that all the energy of the emitted photons an be onverted into

ionization. The ionizationhistory onstrains ALP lifetimes muh longer than the

age of the universe,

τ & 10 ²⁴

^s, ^whih ^means ^that ^only ^less ^than ^one ^ALP ^out ^of

ten millions an deay. The eet of the deay of a large population of partiles

has atastrophi eets, but only extremely small perturbations to the standard

osmologial senarioare allowed.

atrestinthe omovingframe. Thespetraluxof photons produedinthe deay

of adiuse pseudosalarpopulationis [31,137℄

dF E

dEdΩ = 1 2π

Γ H(z)

n φ (z)

(1 + z) ³ =

^(3.15a)

≃ n ¯ φ0

2πτ H 0

E 0

m φ /2 3/2

exp

"

− t 0

τ

E 0

m φ /2 3/2 #

,

^(3.15b)

wherethesubsript 0meansquantitiesatpresenttime,

n ¯ φ0

^is^the^putative^number

density if

φ

^would ^be ^stable, ^and

E 0

^is ^the ^energy ^at ^whih ^the ^photon ^would ^be

seen today. The photon initialenergy is

m φ /2

^and ^thus ^the ^redshift ^of ^the ^deay

is given by

1 + z = (m φ /2)/E 0

^. ^F^or ^simpliity ^we ^assumed ^matter ^domination

negleting

O (1)

^orretions^due^to^the^osmologial^onstant. ^T^o^take^into^aount

theopaityofthe universetotheultravioletradiation,theuxhastobeorreted

multiplyingit by the survival probability

P (z) = e ^−κ(z,E ⁾

^(3.16)

κ(z, E) = Z z

0 dz ^′

H(z ^′ )(1 + z ^′ ) n H (z ^′ )σ ph

ion (E) ,

^(3.17)

where

E = E 0 (1+z)

n H

^is^the^hydrogen^number^density^and

σ ph

ion

^is^the^hydrogen

photoeletriross-setiongivenbyequation(3.14),with

E _1s = 13.6 eV

^and

Z = 1

The ontributionfrom helium photoionizationis not relevant. One weorreted

thespetrumbythisfator,weompareditwiththeextragalatibakgroundlight

(EBL) spetrum reviewed in [40℄. We orreted also the mirowave range of the

spetrumsubtrating the CMB ontribution, whihis known extremely wellafter

the measurement of COBE-FIRAS. The EBL spetrum does not present sharp

features that eventually ould be reognised to be the ontribution of a deaying

partile. Therefore we require the deay photon spetrum to be lower than the

EBL. The region of parameter spae exluded by this omparison is plotted in

dullgreen ingure3.3, whereitis labelledEBL.The sameapproahwas already

used in referenes [39,40℄ to look for axions in the optial EBL. In the part of

parameter spae that an be onstrained in this way, the ALP abundane would

be the minimum onsidered

n φ /n γ = (3.9/106.75)/2 ≃ 0.019

^. ^Sine ^the ^thermal

ALPswouldprovidetoomuhDM for

m _φ > 154

^eV,ⁱⁿ^our^analysis^we^found^more

onvenient toassume that ALPs provide theright amount of DM above this mass,

while below it the standard thermal abundane is onsidered. In this way, we

avoid todupliate the limitsin the region whih is already exluded, and provide

a bound that only implies ALPs to onstitute the DM. Of ourse this requires

some non-standard dilution of the ALP number density by additional

degrees-of-freedom above the eletroweak sale. The bound gets severely degraded in the

m φ = 13.6

300

^e^V ^range, ^where ^not ^only ^absorption ^is ^very ^strong ^but ^also ^the

experimental data are extremely hallenging and the EBL spetrum has only an

upperbound estimate. Of oursethese fats are losely related.

Inordertogainsensitivity,amoreeetivestrategytosearhfordeayphotons

is to examine the light emitted in galaxies and large sale strutures. There, the

dark matter density is above the average, providing therefore an enhaned signal

for the deay produts. However, also the bakgrounds are above the EBL, but

anyway the deay photons should stand above the bakground appearing like a

peak. Searhes of axions in the visible have been presented in [38,39℄ and more

reently in [41℄. The limits from these referenes are plotted in gure 1.2 in the

blue bandlabelledTelesope. In[138,139℄ sterileneutrino darkmatterdeaysin

a photonplus anativeneutrino were searhed inX-rays with the samemethods.

Wefollowedthesereferenes,resalingtheresultsfortheALPdeaymodewhen

exploiting neutrino searhes, and abundane when the thermal ALP yield gives

Ω _φ < Ω _DM

^. ^The ^exlusion ^bounds ^we ^obtained ^are ^plotted ⁱⁿ ^gure ^3.3, ^labelled

respetively Optial and X-Rays. We have also plotted in gure 3.3 the limit

obtained by galatilinesearhes inthe

γ

^-ray ^range ^for^deaying^DM, ^labelling ^it

γ

^-rays. ^W^e^took^the^data^from^[140℄^and^again^we^resaled ^them^for^the^ALP^deay

mode. Anyway, onsidering that the region onstrained by

γ

^-ray observations is well above the

T fo > m Pl

^line, ^it ^makes ^probably ^not ^too ^muh ^sense ⁱⁿ ^our

thermal-ALP piture, but we showed it anyway for the sake of ompleteness. In

this

γ

^-ray ^region, ^a ^very ^suggestive ^yet ^tentative ^laim ^of ^a ^detetion ^at

ω ≃ 130

GeV appeared reently [141℄. Unfortunately, we an not relate it to deaying

pseudosalars, sine itan only be attributed toannihilating DM.

In these spetral analyses, the monohromati emission line aused by the

deay or annihilation is reonstruted, and its intensity traks the density

DM

EBL

X-Rays

Γ-Rays

Optical

CMB Μ CMB y x _ion

x ion

HB excluded

KSVZ axion

T fo

= E

EW

T fo = m

Pl

Τ=10 ¹⁷ s

0 2 4 6 8 10

10 15 20 25 30

Log ₁₀ m _Φ @eVD Log 10 Τ @ s D

Figure 3.3: Summarising pitureforthis hapterinthe

m _φ

τ

^parameter ^spae.

All the boundsare desribed in thetext. In light green are plotted the bounds

relatedto CMB distortion,respetively labelled CMB y,CMB

µ

^and

x ion

^. ^In

dull green are the limits from the lak of diret observation of deay photons.

They are EBL and Optial, X-Ray and

γ

^-Ray. ^W^e ^onsidered ^the ^ALP

abundane provided by the thermal prodution mehanism for all the bounds

in all the parameter spae but diret deay photon detetion in the

m _φ > 154

eV range. Here, ALPs would be overprodued with respet to the measured

DM abundane,therefore

Ω _DM

^is^taken ^for^ALP ^abundaneⁱⁿ^the^diret^photon

detetion limits. The orange region labelled DM overs the exlusion bound

through lensing observations. The highresolution of modern instrumentspermits

to extrat dierent spetra from dierent zones of an objet. The emission line

are thus searhed privilegingthe higher density regions.

The radiation produed by ALP deay, being proportional to the reli ALP

density, would be linearly sensitive to the partile ontent above the EW sale

through

g ∗ (T fo )

^. ^The ^onstraint ^on

g φ

^relaxes ^as

p 1 + new dof/106.75

^with ^the

new thermal degrees of freedom with masses between the EW and

T fo

ⁱⁿ ^the

m φ < 154

^e^V ^range. ^The ^new ^degrees ^of ^freedom ^do ^not ^aet ^the ^diret ^deay

photon detetion bound for

m φ > 154

^e^V, ^sine ⁱⁿ ^this ^mass ^range ^we ^assumed

Ω _φ = Ω _DM

Diluting neutrinos and nuleons:

limits from dark radiation and big-bang

nuleosynthesis

Ifthepseudosalardeayhappenslongbeforematter-radiationdeoupling,the

pri-mordialplasmahas enoughtime toregain thermodynamiequilibrium. Thisdoes

not meanthatisnot possibletoobserveanyonsequene ofthis event. Thedeay

modiesthe photonabundane,and leaves unalteredthe baryonandthe neutrino

abundanesif neutrinoshavealreadydeoupled fromthe bath. The

baryon-to-photon ratio

η

^and ^the ^eetive ^number ^of ^neutrinos

N _eff ∝ ρ _ν /ρ _γ

^are ^eetively

diluted, and they an bemeasured through the analysis of CMB anisotropies and

the observation of primordialelementsabundane.

Afast estimateofthe entropy inrementdue tothe deay is auseful guideline

to understand the topis of this hapter. Sine neutrinos are not aeted by the

deayif alreadydeoupled, itprovesonvenient tonormaliseabundanesinterms

of the neutrino temperature whih redshifts as

T ν ∝ R ⁻¹

^. ^So ^we ^dene

T ≡

T γ

T ν

3 ∝ n γ

n ν

and

ξ ≡ T φ

T γ

3 = n φ

n ^eq _φ .

^(4.1)

The initialondition we use is

T γ,0 = T ν,0 = T φ,0 = 2

^Me^V, ^when ^neutrinos ^have

deoupled and

e ⁺ e ⁻

annihilation has not yet begun. A rst stage is dened by

T _γ,1 ≃ m _e /10

^, ^when eletron-positron annihilation is over. Entropy onservation

during

e ⁺ e ⁻

annihilationimplies

T ⁰ 7

2 + 2 + ξ 0

= T ¹ (2 + ξ 1 ) ,

^(4.2)

where

7/2

^and ² ^are ^the

e ⁺ e ⁻

^and

γ

^entropy ^degrees ^of ^freedom. ^Beause ^of ^the

initialondition we have hosen,

T 0

^and

ξ 0

^are ^equal^to ^1. Considering the axion ase,fromgure2.11 wesee thatwe aredesribing anepohfarbelowthe dashed

line,whihmeansthattheyarerelativistiinthemassrangeweareinterested. We

assumekineti equilibrium,even if axions deouple during the

e ⁺ e ⁻

annihilation.

If axions or ALPs deouple before

e ⁺ e ⁻

annihilation, like neutrinos, they are not aeted by the entropy released in the annihilation. Photons are heated by

the standard amount, thus

T 1 = 11/4

^, ^while^the ^produt

ξ T

^is^onserved. ^On ^the

other hand, if they deouple during or after

e ⁺ e ⁻

annihilation they are heated, sharingsome of the eletron entropy, and thus we have

T 1 = 13

2(ξ 1 + 2) .

^(4.3)

Ifpseudosalars are fully oupled during

e ⁺ e ⁻

annihilation,then

ξ 1 = 1

^and

T ¹ = 13/6

^. ^In ^general ^we ^have

11/4 < T ¹ < 13/6

^. ^Figure ^2.12 ^plots ^the ^value ^of

ξ

^for

the axionat

T 1

The eventual deay of a population of pseudosalars makes the entropy they

arry tobe transferred to the photon bath. In the ase that

φ

^never ^leaves

ther-mal equilibrium, we easily nd that the nal photon abundane is

13/11

^times

the standard value. In the axionase, we have pointed out that this happens for

m a >

∼ 20

^ke^V. ^In ^general^the ^entropy ^transfer ^to ^photons ^depends ^on^two

parame-ters, the initialpseudosalarabundane, parametrisedby

ξ

^,^and ^the ^eetiveness

of inverse deay whenaxions and ALPs beome non-relativisti.

Ifthe deay freezes-inwhen

T γ > m φ /3

^, ^thepseudosalar temperatureathes up with the photon one whihhas toderease beause of energy onservation. In

this proess, radiationis simply shued from one form toanother and omoving

energy remains onserved. Using again neutrinos as a ruler, in our units the

energydensityis

ρ γ+φ /ρ ν ∝ (2 + ξ 1 4/3 ) T 1 ^4/3

^before^the ^reoupling. ^After^reoupling

T ∗

^, ^the pseudosalar temperature ath up with the photonone, thus we have

ρ _γ+φ /ρ _ν ∝ (2+1) T ^∗ ^4/3

^,^and,^beause^of^energyonservation,thephotonabundane

is redued to

T ^∗ = T ¹ 2 + ξ 1 4/3

2 + 1

! 3/4

.

^(4.4)

Comoving entropy inreases by the fator

[(2 + 1)/(2 + ξ 1 )] T ^∗ / T ¹

^. ^At ^the ^later

temperature

T 2

^, ^when pseudosalars beome nonrelativisti and their abundane gets Boltzmann suppressed, the deay transfers the pseudosalar entropy to the

photon bath,heating it aording to

2 T 2 = (2 + 1) T ∗ .

^(4.5)

Putting all together, the nal photon heating by pseudosalars reoupling and

adiabatideay is

S 2

S 1

= T 2

T ¹ = 3 2

2 + ξ ₁ ^4/3 3

! 3/4

,

^(4.6)

whihis alsothe ratio ofthe entropy density afterand beforethe deay. Werefer

to this ase as equilibrium deay, sine it happens in a loal thermal equilibrium

ondition.

If pseudosalars reouple non-relativistially and they dominate the energy

budget of the universe just before reoupling, their out-of-equilibrium deay has

more dramati onsequenes. The out-of-equilibrium deay an produe a vast

inrease in the entropy density, beause the energy whih is rst stored in the

mass of a non-relativisti partile is suddenly released as radiation. This ase is

illustrated by an analyti approximation tothe entropy generation [76℄

S ₂ S 1

= T 2

T 1

= 1.83 h g _∗S ^1/3 i ^3/4 m _φ Y _φ (T ₁ ) r τ

m Pl

,

^(4.7)

where

h g _∗S ^1/3 i

^denotes ^an^average ^over^the ^deay ^time ^and

Y φ ≡ n φ /s

^. ^This ^formula

is valid if the deay produes relativisti partiles, and the energy densities of

all the speies other than

φ

^are ^negligible ^before ^the ^deay^. ^The ^energy ^density

sales as

ρ ∝ R ⁻³

^for non-relativisti speies, while

ρ ∝ R ⁻⁴

^for ^radiation. ^A

pseudosalardominateduniverserequiresnon-relativistipartileswhihare

long-livedenoughtosurviveuntilthepseudosalar-radiationequality. Thisonditionis

1 10 10 ² 0

0.5 1 1.5 2 2.5 3 3.5 4

m a @keVD S 2 S 1

Figure 4.1: Photon densityinrease inour modied osmologyasexpressedby

S ₂ /S ₁

^for

C _γ = 1.9

^. ^Solid ^and ^dashed^lines^stand ^for ^hadroni ^andnon-hadroni

axions respetively. In the latter ase

C _e = 1/6

^. ^The ^thin ^blue ^lines ^show ^the

valueifweassume thatnoentropyisgenerated inaxion deay. Thevertialline

signals theendof

e ⁺ e ⁻

annihilationepoh.

heavy pseudosalars deay early beause of the

m ⁻³ _φ

^dependene ^of ^the ^lifetime,

while very lightpartiles beomes not-relativistilate.

In gure 4.1 we show using blak lines the resulting

S ₂ /S ₁

^from ^a ^numerial

solutionof the set of Boltzmann equations for the axion phase spae distribution

and theneutrino, photonand eletron abundanesas afuntionof

m a

ⁱⁿ^the ^ase

C γ = 1.9

^, ^with ^and ^without ^the ^eletron ^oupling. ^The ^blue ^thin ^lines ^represent

just the inrease in the entropy density due to reshuing of entropy between

axions and photons, whih follows equation (4.6), ignoring the entropy generated

in the out-of-equilibrium deay. It is interesting to observe how dramati is the

eetof theaxiondominationonthe omovingentropydensity. Startingfromthe

very right, in the high mass range, we an observe how negligibleis the eet of

the deay, sine in this ase the axion population has already disappeared from

the bath when

T = T 1

^. ^The ^value ^of

S 2 /S 1

^inreases ^moving ^to ^the ^left, ^as ^the

The maximum value allowed in this situation is provided by equation (4.6) and

3/2

^. ^Continuing ^to ^the ^left, ^espeially ^if ^no ^diret ^oupling ^with ^the ^eletron

is present,axions experienea periodof deoupling followed by reouplingdue to

inversedeay. Thevalueof

S 2 /S 1

^dereases^beause

ξ 1

^isinreasinglylessthanone in equation(4.6). Finally, the out-of-equilibriumrange begins,and the blak and

blue lines diverge. Here,

ξ 1

^tends ^to^zero, ^and ^indeed^the ^blue ^lines ^approah^two.

But the atual value of

S 2 /S 1

^is ^aeted ^by ^entropy generation, and it is highly boosted if the deay happens late enoughfor the axiontodominate the universe.

Again, the additionalouplingto eletrons makesthe transitionfromequilibrium

to non-equilibriumdeay shifttowards lower mass.

Im Dokument Cosmological limits on axions and axion-like particles (Seite 77-87)

m φ = 100

10 22

ω

σ ph

ion ∼ 256π 3

α em

Z 2

E 1s (Z) ω

7/2

a 2 0 ,

Z

E 1s = 13.6Z 2

1s

a 0 = (αm e ) −1 = 5.292 × 10 −9

ω

z = 6

100

τ 6

x ion

m φ . 300

τ & 10 24

dF E

dEdΩ = 1 2π

Γ H(z)

n φ (z)

(1 + z) 3 =

≃ n ¯ φ0

2πτ H 0

E 0

m φ /2 3/2

exp

"

− t 0

τ

E 0

m φ /2 3/2 #

,

n ¯ φ0

φ

E 0

m φ /2

1 + z = (m φ /2)/E 0

O (1)

P (z) = e −κ(z,E )

κ(z, E) = Z z

0

dz ′

H(z ′ )(1 + z ′ ) n H (z ′ )σ ph

ion (E) ,

E = E 0 (1+z)

n H

σ ph

ion

E 1s = 13.6 eV

Z = 1

n φ /n γ = (3.9/106.75)/2 ≃ 0.019

m φ > 154

m φ = 13.6

300

Ω φ < Ω DM

γ

γ

γ

T fo > m Pl

γ

ω ≃ 130

DM

EBL

EBL

X-Rays

Γ-Rays

Optical

CMB Μ CMB y x ion

x ion

HB excluded

KSVZ axion

T fo

= E

EW

10 ²²

Z ²

a ² ₀ ,

E 1s = 13.6Z ²

a 0 = (αm e ) ⁻¹ = 5.292 × 10 ⁻⁹

x _ion

τ & 10 ²⁴

(1 + z) ³ =

P (z) = e ^−κ(z,E ⁾

dz ^′

H(z ^′ )(1 + z ^′ ) n H (z ^′ )σ ph

E _1s = 13.6 eV

m _φ > 154

Ω _φ < Ω _DM

CMB Μ CMB y x _ion

Τ=10 ¹⁷ s

Log ₁₀ m _Φ @eVD Log 10 Τ @ s D

m _φ

m _φ > 154

Ω _DM

Ω _φ = Ω _DM

N _eff ∝ ρ _ν /ρ _γ

T ν ∝ R ⁻¹

n ^eq _φ .

e ⁺ e ⁻

T _γ,1 ≃ m _e /10

e ⁺ e ⁻

T ⁰ 7

= T ¹ (2 + ξ 1 ) ,

e ⁺ e ⁻

e ⁺ e ⁻

e ⁺ e ⁻

e ⁺ e ⁻

e ⁺ e ⁻

T ¹ = 13/6

11/4 < T ¹ < 13/6

ρ γ+φ /ρ ν ∝ (2 + ξ 1 4/3 ) T 1 ^4/3

ρ _γ+φ /ρ _ν ∝ (2+1) T ^∗ ^4/3

T ^∗ = T ¹ 2 + ξ 1 4/3