Axion Cold Dark Matter in Standard and Non-Standard Cosmologies

(1)

Paolo Gondolo

University of Utah

Axion Cold Dark Matter in Standard and Non-Standard Cosmologies

Visinelli, Gondolo, arxiv:0903.4377, Phys. Rev. D 80, 035024 (2009) Visinelli, Gondolo, arxiv:0912.0015

Luca Visinelli

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Axion cold dark matter

Study axion parameter space imposing Ω

a

=Ω

CDM

=0.1131+0.0034

When are axions 100% of cold dark matter?

And update cosmological constraints and include anharmonicities

(3)

Axions as solution to the strong CP problem

The strong CP problem

Vacuum potentials A_µ = iΩ∂_µΩ⁻¹ with _{Ω → e}^2πin as r → ∞ Vacuum state |θ� = �

n e^−inθ |0�

New term in lagrangian L^θ = θ _32π^g²2 F_a^µν F˜_aµν

violates P and T but conserves C, thus produces a neutron electric dipole moment d_n ≈ e(m^q/M_n²)θ L^θ

Experimentally d_n < 1.1 × 10⁻²⁶ ecm so θ < 10⁻⁹–10⁻¹⁰ Why θ should be so small is the strong CP problem

(4)

Axions as solution to the strong CP problem

The Peccei-Quinn solution

New lagrangian _La = −¹₂∂^µa∂_µa + _f^a

a

g²

32π² F_a^µνF_aµν + L^int(a)

Before QCD phase transition,�θ� can be anything Introducing a U(1)PQ symmetry replaces

θ_total = θ + arg det M_quark

static CP-violating angle dynamic CP-conserving field

θ(x) = a(x)/f_a

axion

V (θ) = m²_af_a²(1 − cos θ)

After QCD phase transition, instanton effects generate and _{�θ� = 0} dynamically

! V(!)

(5)

Axions as dark matter

Hot

Cold

Produced thermally in early universe

Important for ma>0.1eV (fa<10⁸), mostly excluded by astrophysics

Produced by coherent field oscillations around mimimum of V(θ)

(Vacuum realignment)

Produced by decay of topological defects

(Axionic string decays)

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Axion cold dark matter parameter space

fa Peccei-Quinn symmetry breaking scale N Peccei-Quinn color anomaly

Nd Number of degenerate QCD vacua

Kim-Shifman-Vainshtain-Zakharov

Dine-Fischler-Srednicki-Zhitnistki Couplings to quarks, leptons, and photons HI Expansion rate at end of inflation

θi Initial misalignment angle

Harari-Sikivie-Hagmann-Chang

Davis-Battye-Shellard Axionic string parameters

Assume N = Nd = 1 and show results for KSVZ and HSHC string network Thus 3 free parameters fa, θi, HI and one constraint Ωa=ΩCDM

(7)

Cold axion production in cosmology

•

Initial misalignment angle θi

•

Coherent axion oscillations start at temperature T1

3H(T1)=m(T1)

•

Density at T1 is

•

Conservation of comoving axion number gives present density Ωa

Hubble expansion parameter non-standard expansion histories differ in the function H(T)

T-dependent axion mass axions acquire mass through

instanton effects at T < Λ ≈ Λ^QCD

Vacuum realignment

n_a(T₁) = ¹₂m_a(T₁)f_a²χ�θi²f (θ_i)�

Anharmonicity correction f (θ)

axion field equation has anharmonic terms ¨θ+ 3H(T) ˙θ + m²_a(T ) sin θ = 0

(8)

Cold axion production in cosmology

•

Energy density ratio (string decay/misalignment)

(String stretching rate)^-2

Axionic string decays

α ≡ ρ^str_a

ρ^mis_a = ξ¯rN_d² ζ

Density enhancement from string decays

Uncertainty in axion spectrum

Fast-oscillating strings (Harari-Hagmann-Chang-Sikivie) Slow oscillating strings (Davis-Battye-Shellard)

¯r = _3β¹^−β₋₁ 0.8

¯r = _3β¹^−β₋₁ ln(t₁/δ)

with a(t)∝^t^β

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Standard cosmology

ρ∝a^-3 ρ∝a^-4

MD RD

Inflation ρ∝V^(φ)

H

²

= 8 π

3 M

_Pl²

ρ

H∝a

^-2

H∝V

^1/2

H∝a

^-3/2

ln a ln H ^Inflation _Reheating

Radiation dominated

Matter dominated

H∝Λ

^1/2

Λ dominated

ρ∝Λ

ΛD

Axion production

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Axion CDM - Standard cosmology

White dwarfs cooling time

Tensormodes

Θ_i"1 Θ_i"0.1

Θ_i"3.14 Θ_i"0.01 Θ_i"0.001 Θ_i"0.0001

#_a $ #_CDM

#_a % #_CDM

ADMXI ADMXII

CARRACK

PLANCK

f^a" T^GH

Axion isocurvature fluctuations

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!3 10^!6 10^!9 10^!12

H_I !GeV"

f a!GeV" m a!eV"

(11)

Axion CDM - Standard cosmology

Tensormodes

Θ_i"1 Θ_i"0.1

Θ_i"3.14 Θ_i"0.01 Θ_i"0.001 Θ_i"0.0001

#_a $ #_CDM

#_a % #_CDM

CARRACK

PLANCK

f^a" T^GH

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!3 10^!6 10^!9 10^!12

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

metry breaks after end of inflation PQ sym

metry breaks during inflation

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Axion CDM - Standard cosmology

Tensormodes

Θ_i"1 Θ_i"0.1

Θ_i"3.14 Θ_i"0.01 Θ_i"0.001 Θ_i"0.0001

#_a $ #_CDM

#_a % #_CDM

CARRACK

PLANCK

f^a" T^GH

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!3 10^!6 10^!9 10^!12

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

metry breaks after end of inflation Axion oscillations start

during inflation

PQ symmetry breaks after end of inflation

•

^{Average θ}ⁱ over Hubble volume

•

Anharmonicities are important

!"#$%&'()*+'$$$$$$$

! !"_"#$%!"#! # &^$

&^$'""$ #(^%^)&

! !"_"#

"_')&

!"#$%#&'()*

+,-.&'((/

θ/π

f (θ) = �

ln _π^eπ2−θ²²

�7/6

•

String decay contribution is

~16% of vacuum realignment

(13)

Axion CDM - Standard cosmology

Tensormodes

Θ_i"1 Θ_i"0.1

Θ_i"3.14 Θ_i"0.01 Θ_i"0.001 Θ_i"0.0001

#_a $ #_CDM

#_a % #_CDM

CARRACK

PLANCK

f^a" T^GH

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!3 10^!6 10^!9 10^!12

H_I !GeV"

f a!GeV" m a!eV"

Axion oscillations start after end of inflation PQ sym

!_a " !_CDM

!_a # !_CDM

10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 0.0

0.2 0.4 0.6 0.8

1.0 10^!³ 10^!⁴ 10^!⁵ 10^!⁶ 10^!⁷ 10^!⁸ 10^!⁹

f GeV"

Θi#Π

ma !eV"

•

Single value of θi throughout Hubble volume

Strong CP problem?

•

Constrained by non-adiabatic fluctuations

PQ symmetry breaks during inflation

Anharmonicities

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Axion CDM - Standard cosmology

Tensormodes

Θ_i"1 Θ_i"0.1

Θ_i"3.14 Θ_i"0.01 Θ_i"0.001 Θ_i"0.0001

#_a $ #_CDM

#_a % #_CDM

CARRACK

PLANCK

f^a" T^GH

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!3 10^!6 10^!9 10^!12

H_I !GeV"

f a!GeV" m a!eV"

ma = 85±3 µeV

(15)

Standard cosmology

ρ∝a^-3 ρ∝a^-4

MD RD

H

²

= 8 π

3 M

_Pl²

ρ

H∝a

^-2

H∝V

^1/2

H∝a

^-3/2

ln a ln H ^Inflation _Reheating

Radiation dominated

Matter dominated

H∝Λ

^1/2

Λ dominated

ρ∝Λ

ΛD

Axion production

(16)

Non-standard cosmology

ρ∝a^-3 ρ∝a^-4

MD RD

H

²

= 8 π

3 M

_Pl²

ρ

H∝a

^-2

H∝V

^1/2

H∝a

^-3/2

H∝a

^-3/2

ln a ln H ^Inflation Reheating

Radiation dominated

Matter dominated

H∝Λ

^1/2

Λ dominated

ρ∝Λ

ΛD

Big-Bang Nucleosynthesis

Axion production

(17)

Low Temperature Reheating cosmology

ρ∝a^-3 ρ∝a^-4

MD RD

H

²

= 8 π

3 M

_Pl²

ρ

H∝a

^-2

H∝V

^1/2

H∝a

^-3/2

H∝a

^-3/2

Radiation dominated

Matter dominated

H∝Λ

^1/2

Λ dominated

ρ∝Λ

ΛD

Axion production

Dominated by the decay of a frozen scalar field

Turner 1983, Scherrer, Turner 1983, Dine, Fischler 1983

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Axion CDM - Low Temp. Reheating cosmology

White Dwarfs Cooling Time

TensorModes

f^a! H ^I!2Π

Axion Isocurvature Fluctuations

Standard TRH ! 4MeV TRH ! 15MeV TRH ! 150MeV

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹

H_I !GeV"

f a!GeV" m a!eV"

(19)

Axion CDM - Low Temp. Reheating cosmology

TensorModes

f^a! H ^I!2Π

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

(20)

Axion CDM - Low Temp. Reheating cosmology

TensorModes

f^a! H ^I!2Π

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

lower TRH

•

^{As T}^RH decreases, fa must increase and ma decrease

White Dwarfs Cooling Time Tensor Modes

BigBangNucleosynthesis

faLTR

ADMX

fastd

TRH!T1std

f_a"H_I!2Π

5 10 50 100 500 1000

10⁸ 10¹⁰ 10¹² 10¹⁴

10^!² 10^!⁴ 10^!⁶ 10^!⁸

TRH!MeV"

fa!GeV" ma!eV"

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Axion CDM - Low Temp. Reheating cosmology

TensorModes

f^a! H ^I!2Π

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

•

^{As T}^RH decreases, constraints from non-adiabatic fluctuations become weaker

•

And the initial misalignment angle θi must be larger

!_a" !_CDM

!_a# !_CDM

f_a"H_I!2Π

WhiteDwarfsCoolingTime

10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 0.0

0.2 0.4 0.6 0.8 1.0

10^!110^!210^!310^!410^!510^!610^!710^!810^!910^!10

Θi#Π

ma !eV"

lower TRH

(22)

Non-standard cosmology

ρ∝a^-3 ρ∝a^-4

MD RD

H

²

= 8 π

3 M

_Pl²

ρ

H∝a

^-2

H∝V

^1/2

H∝a

^-3/2

H∝a

^-3/2

Radiation dominated

Matter dominated

H∝Λ

^1/2

Λ dominated

ρ∝Λ

ΛD

Big-Bang Nucleosynthesis

Axion production

(23)

Kination cosmology

ρ∝a^-3 ρ∝a^-4

MD RD

H

²

= 8 π

3 M

_Pl²

ρ

H∝a

^-2

H∝V

^1/2

H∝a

^-3/2

H∝a

^-3

ln a ln H ^Inflation ^Kination

Radiation dominated

Matter dominated

H∝Λ

^1/2

Λ dominated

ρ∝Λ

ΛD

Axion production

Kination period dominated by kinetic energy of scalar field

Ford 1987

(24)

Axion CDM - Kination cosmology

TensorModes

f^a!H ^I!2Π

Standard Tkin ! 4MeV Tkin ! 300MeV Tkin ! 700MeV

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹ 10^!¹²

H_I !GeV"

f a!GeV" m a!eV"

(25)

Axion CDM - Kination cosmology

TensorModes

f^a!H ^I!2Π

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹ 10^!¹²

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

(26)

Axion CDM - Kination cosmology

TensorModes

f^a!H ^I!2Π

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹ 10^!¹²

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

•

^{As T}^kin decreases, fa must decrease and ma increase

White Dwarfs Cooling Time Tensor Modes

BigBangNucleosynthesis

fastd

ADMX

fakin stdT!Tkin1

f_a"H_I!2Π

5 10 50 100 500 1000

10⁸ 10¹⁰ 10¹² 10¹⁴

10^!² 10^!⁴ 10^!⁶ 10^!⁸

Tkin!MeV"

fa!GeV" ma!eV"

lower Tkin

•

String decay contribution is 15 × vacuum realignment

(27)

Axion CDM - Kination cosmology

TensorModes

f^a!H ^I!2Π

ADMX

10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10⁸

10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

10^!³ 10^!⁶ 10^!⁹ 10^!¹²

H_I !GeV"

f a!GeV" m a!eV"

PQ sym

•

^{As T}^kin decreases, constraints from non-adiabatic fluctuations become stronger

•

And the initial misalignment angle θi must be smaller

lower Tkin

!_a" !_CDM

!_a# !_CDM

f_a"H_I!2Π

WhiteDwarfsCoolingTime

10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 0.0

0.2 0.4 0.6 0.8 1.0

10^!110^!210^!310^!410^!510^!610^!710^!810^!910^!10

Θi#Π

ma !eV"

lower Tkin

(28)

Conclusions

• If the Peccei-Quinn symmetry breaks after inflation ends,

the axion mass must be m

a

=85±3 µeV in standard cosmology

• If the Peccei-Quinn symmetry breaks during inflation,

cosmological limits on non-adiabatic fluctuations constrain parameter space and a specific initial misalignment angle θ

i

must be chosen

For axions to be 100% of cold dark matter...

-

much smaller ma in LTR cosmology

-

much larger ma in kination cosmology

-

larger allowed region and larger θⁱ in LTR cosmology

-

smaller allowed region and smaller θⁱ in kination cosmology