Ionization history of the universe

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.

in their dierent energy levels. The result of this alulation is usually given in

terms of the fration of free eletrons, or ionization fration

x ion

^, ^as ^a ^funtion

of redshift. The reombination of the hydrogen freezes out at

z ∼ 800

^beause

of the expansion of the universe, leaving a residual ionization fration of order

x ion ∼ O (10 ⁻⁴ )

^. ^The ^universe ^fully ^ionizes ^again ^muh ^later, ^between ^redshifts

6

and

10

^,^presumably ^due^toultravioletemissionfromtherst galaxies. Thedetails of the reionization proess are still not wellunderstood[133℄.

Theslight imprint thatthe free eletrons leavein the CMB through Thomson

sattering, forinstane in the polarization,givesusinformationabout the history

of reombination and reionization. The optial depth for CMB photons is one of

the parameters that an be measured from the CMB multipoleanalysis and it is

dened tobe

τ _opt (z ₁ , z ₂ ) = − Z z 2

z 1

σ T n e (z)x ion (z)

H(z)(1 + z) dz ,

^(3.11)

where allthe quantities inthe integral are expressed as afuntion of redshift.

TheWMAP7measured

τ opt

^after^deoupling^to^be

0.088 ± 0.015

^[86℄. ^A ^fator

0.04

0.05

^of^this ^an ^be ^attributed^to^a^fully ^ionized^universe ^up ^to^about ^redshift

6, whih is supported by the absene of Ly-

α

^features ⁱⁿ ^quasar ^spetra. ^The

origin of the remaining fration,

τ ₆ = τ _opt (6, z _dec ) ≃ 0.04

^, ^is ^still ^unertain ^and

leaves somespae for ALP deay.

The photons produed by ALP deay after deoupling are free to propagate

sine there are almost no free eletrons to interat with. However, ultraviolet

ra-diationan interatwithatomsandphotoionizethem. Theuniverseisindeedvery

opaque to ultraviolet radiation. This prevents us from deteting deay photons

in the

13.6

300

^e^V ^range, ^but ^it ^does ^not ^mean ^that ^we ^an ^not ^onstrain ^them,

sine the photoionisation triggers an inreasing of the ionization fration of the

universe. This argumentdoesnot hold for axions beauseinthis mass range they

deaybeforereombination. Ifweassumethateahdeay photonionizesonlyone

H atom immediately after the ALP deay, whih is a rst rough approximation,

the numberof ionizations per unit time an be estimated to be[134℄

ξ(z) ∼ 2 τ

n _φ (z)

n H (z) ∼ 2 × 10 ⁻³ m _φ 100 eV

3 g _φ 10 ⁻¹³ GeV ⁻¹

2 e ⁻ ² ³ ^H(z)τ ¹ Myr ⁻¹ .

^(3.12)

10 20 50 100 200 500 1000 2000 0.001

0.01 0.1 1

z x e

m Φ =100 eV 10 ₁₈

10 ₂₀ s 10 s

22 s 10 24

s 10 ₂₄

s

10 20 50 100 200 500 1000 2000

0.001 0.01 0.1 1

z x e

Τ=10 ²² s 50

eV 100

eV 500 1000 eV

eV

Figure 3.2: The eet of ALP deay on the ionization fration, plotted in

funtion of the reshift

z

^. ^The ^standard ^ase ^with ^no ^deay ^is ^plotted ^with ^the

solidline. Left: deayofa

100

^eV^mass^ALP ^with

τ = 10 ¹⁸

^s,

10 ²⁰

^s,

10 ²²

^s,

10 ²⁴

s and

10 ²⁶

^s. ^Right: ^deay^of ^a

τ = 10 ²²

^s ^ALP ^with

m _φ = 1000

^e^V,

500

^eV,

100

eV and

50

^eV.

Ifwe now multiplythis quantity by atypialtime sale [134℄

t _H = 1/H (z) ∼ 2.4 Myr [501/(1 + z)] ^3/2 ,

^(3.13)

wherewehavenegleted

Ω r

^and

Ω Λ

ⁱⁿ

H(z)

^,^we^get ^aonservativeestimateofthe degreeofionizationinduedbytheALPdeaysuptoaertainredshift. ALPswith

100eVmassand

g _φ ∼ 10 ⁻¹³

^Ge^V

⁻¹

^would^have^produed^an^ionization^omparable

with the standard residual value

10 ⁻⁴

^already ^at ^high ^redshifts,

z ∼ 500

^. ^Sine

the ALP-indued ionization grows in time as

(1 + z) ^−3/2

^, ^it ^shows ^a potentially interesting eet. We ould even think that ALPs lose tothese parameters may

be responsible for the full reionisation of the universe. However, the

(1 + z) ^−3/2

dependene is too soft reionization seems to be a muh more abrupt proess,

usually parametrised to be almost a step funtion in

x ion (z)

^and ^we ^an ^only

hek if ALPs providea

τ 6

^ompatible^with observations.

Inordertoobtainamoredetailedonstraint,wehavealulatedthe ionization

historyoftheuniverseinthedeayingALPosmology[2,3℄byintroduingtheALP

ionizations in the reombination ode RECFAST [135℄. In gure 3.2, we plot the

ionizationhistoryoftheuniverseinfuntionof

z

^for^several ^examples^of^ALP^mass

andlifetime. Weset theALPfrationtotoaountforalltheDM,

Ω _DM h ² = 0.11

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.

Im Dokument Cosmological limits on axions and axion-like particles (Seite 74-77)

d µ a

dt = − 2 2.14

3 ρ γ

d ρ a

dt − 4 n γ

dn a

dt

,

d n a /dt = − n a /τ

d ρ a /dt = m a d n a /dt

µ

m a

δ ≡ C γ /1.9

C γ = 1.9

m a > 8.7 keV

.

µ

m a

δ < 1

C γ

m a

C γ

µ

m a

δ < 0.1

µ

µ

µ

x ion

z ∼ 800

x ion ∼ O (10 −4 )

6

10

τ opt (z 1 , z 2 ) = − Z z 2

z 1

σ T n e (z)x ion (z)

H(z)(1 + z) dz ,

τ opt

0.088 ± 0.015

0.04

0.05

α

τ 6 = τ opt (6, z dec ) ≃ 0.04

13.6

300

ξ(z) ∼ 2 τ

n φ (z)

n H (z) ∼ 2 × 10 −3 m φ 100 eV

3 g φ 10 −13 GeV −1

2

e − 2 3 H(z)τ 1 Myr −1 .

10 20 50 100 200 500 1000 2000 0.001

0.01 0.1 1

z x e

m Φ =100 eV 10 18

10 20 s 10 s

22

s 10 24

s 10 24

s

10 20 50 100 200 500 1000 2000

0.001 0.01 0.1 1

z x e

Τ=10 22 s 50

eV 100

eV 500 1000 eV

eV

z

100

τ = 10 18

10 20

10 22

10 24

10 26

τ = 10 22

m φ = 1000

500

100

50

3 ρ _γ

dt − 4 n _γ

m _a > 8.7 keV

m _a

x ion ∼ O (10 ⁻⁴ )

τ _opt (z ₁ , z ₂ ) = − Z z 2

τ ₆ = τ _opt (6, z _dec ) ≃ 0.04

n _φ (z)

n H (z) ∼ 2 × 10 ⁻³ m _φ 100 eV

3 g _φ 10 ⁻¹³ GeV ⁻¹

e ⁻ ² ³ ^H(z)τ ¹ Myr ⁻¹ .

m Φ =100 eV 10 ₁₈

10 ₂₀ s 10 s

s 10 ₂₄

Τ=10 ²² s 50

τ = 10 ¹⁸

10 ²⁰

10 ²²

10 ²⁴

10 ²⁶

τ = 10 ²²

m _φ = 1000

t _H = 1/H (z) ∼ 2.4 Myr [501/(1 + z)] ^3/2 ,

g _φ ∼ 10 ⁻¹³

⁻¹

10 ⁻⁴

(1 + z) ^−3/2

(1 + z) ^−3/2

Ω _DM h ² = 0.11

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