Axion Cold Dark Matter in Non-Standard Cosmologies

Paolo Gondolo University of Utah

Visinelli, Gondolo, arxiv:0903.4377, Phys. Rev. D 80, 035024 (2009) Visinelli, Gondolo, arxiv:0912.0015, Phys. Rev. D 81, 063508 (2010)

A wider range of masses

Luca Visinelli

Thursday, April 26, 12 Axion cold dark matter

When are axions 100% of cold dark matter?

Study axion parameter space imposing

Ωa=ΩCDM=0.1131+0.0034

Thursday, April 26, 12 Standard cosmology

ΛD ρ∝Λ 2 8π H = 2 ρ MD ρ∝a-3 3MPl RD ρ∝a-4 Inﬂation

H H∝V1/2 Reheating Inﬂation ρ∝V(φ) ln ln Radiation dominated H∝a-2 Matter dominated H∝a-3/2 Λ dominated H∝Λ1/2 ln a Axion production

Thursday, April 26, 12 Non-standard cosmology

ΛD ρ∝Λ 2 8π H = 2 ρ MD ρ∝a-3 3MPl RD ρ∝a-4 Inﬂation H H∝V1/2 Reheating Inﬂation ρ∝V(φ) ln ln H∝a-3/2 Radiation Big-Bang dominated Nucleosynthesis H∝a-2 Matter dominated H∝a-3/2 Λ dominated H∝Λ1/2 ln a Axion production

Thursday, April 26, 12 Non-standard cosmology

Scalar-tensor gravity

Usual metric tensor gµν + scalar ﬁeld φ with action 1 4 ¯2 ¯ ¯ µν ¯ ¯ ¯ ¯ Sg = ! d x√ g¯ φ R +4ω(φ)¯g ∂µφ∂ν φ 4V (φ) 16π − " − # (Jordan frame) For example, Brans-Dicke theory has ω(φ)=ω, V(φ)=0

The Friedmann equation has extra terms

2 8π 1 2 ˙2 H = 2 ρ + 2 MPlφ + V (φ) 3MPl ! " (Einstein frame)

Thursday, April 26, 12 Non-standard cosmology

Braneworlds

4-dimensional spacetime embedded 5-th dimension brane in a higher dimensional spacetime 4-dimensional spacetime For example, Randall-Sundrum model II

Extra term in the Friedmann equation 2 2 2 8π MPlρ H = 2 !ρ + 6 " 3MPl 96πM5

M5 = Planck’s constant in the bulk

Thursday, April 26, 12 Non-standard cosmology

Moduli ﬁelds

In string theory, moduli ﬁelds Calabi-Yau parametrize the shape and manifold 5 5 size of the compactiﬁed extra z1 +z2 =1 dimensions J.-F. Colonna, www.lactamme.polytechnique.fr

Moduli ﬁelds could dominate the energy of the universe at certain times, with some moduli ﬁelds decaying into ordinary and dark matter particles

For example, Moroi-Randall and Acharya-Bobkov-Kane-Kumar-Shao

Thursday, April 26, 12 Non-standard cosmology

Low temperature reheating

A decaying scalar ﬁeld is a simple model of reheating at the end of inﬂation

ρ a-3 ln R a-4 φ Tmax The reheating temperature T -3/8 ln a could be as low as 4 MeV TRH a-1

inflation reheating radiation ln a

Thursday, April 26, 12 Low Temperature Reheating cosmology

ΛD ρ∝Λ 2 8π H = 2 ρ MD ρ∝a-3 3MPl RD ρ∝a-4 Inflation H H∝V1/2 Reheating Inflation ρ∝V(φ) ln ln H∝a-3/2 Radiation dominated Dominated by the H∝a-2 decay of a frozen Matter scalar field dominated H∝a-3/2 Λ dominated H∝Λ1/2 ln a Axion production Turner 1983, Scherrer, Turner 1983, Dine, Fischler 1983

Thursday, April 26, 12 Kination cosmology

ΛD ρ∝Λ 2 8π H = 2 ρ MD ρ∝a-3 3MPl RD ρ∝a-4 Inflation Kination H 1/2 Inflation ρ∝V(φ) H∝V -3 ln ln H∝a Radiation Kination period dominated -2 dominated by H∝a Matter kinetic energy of dominated scalar field H∝a-3/2 Λ dominated H∝Λ1/2 ln a Axion production Ford 1987

Thursday, April 26, 12 Axions as dark matter

Hot

Produced thermally in early universe 8 Important for ma>0.1eV (fa<10 ), mostly excluded by astrophysics

Cold

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

Produced by decay of topological defects (Axionic string decays)

Thursday, April 26, 12 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-Hagmann-Chang-Sikivie Davis-Battye-Shellard Axionic string parameters

Assume N = Nd = 1 and show results for KSVZ and HHCS string network

Thus 3 free parameters fa, θi, HI and one constraint Ωa=ΩCDM

Thursday, April 26, 12 Cold axion production in cosmology

Vacuum realignment

• Initial misalignment angle θi • Coherent axion oscillations start at temperature T1 3H(T1)=m(T1)

Hubble expansion parameter T-dependent axion mass non-standard expansion histories axions acquire mass through differ in the function H(T) instanton effects at T < Λ ≈ ΛQCD

1 2 2 Density at T1 is n (T )= m (T )f ✓ f(✓ ) • a 1 2 a 1 a h i i i Anharmonicity correction f (θ) ¨ ˙ 2 axion ﬁeld equation has anharmonic terms ✓ +3H(T )✓ + ma(T )sin✓ =0 • Conservation of comoving axion number gives present density Ωa

Thursday, April 26, 12 Cold axion production in cosmology

Axionic string decays • Energy density ratio (string decay/misalignment)

(String stretching rate)-2 Density enhancement from string decays str 2 ⇢a ⇠rN¯ d ↵ mis = ⌘ ⇢a ⇣

Uncertainty in axion spectrum

1 Slow-oscillating strings (Davis-Battye-Shellard) r¯ = 3 1 ln(t1/) 1 Fast-oscillating strings (Harari-Hagmann-Chang-Sikivie) r¯ = 3 1 0.8 1 2 ⇠ = 2 3 + (4c + 0)2 12 +4 with a(t)∝tβ 4c2 ⇣ p ⌘ c =(1+2 ⇠std)/(4⇠std)

Thursday, April 26, 12 p Standard cosmology

ΛD ρ∝Λ 2 8π H = 2 ρ MD ρ∝a-3 3MPl RD ρ∝a-4 Inﬂation

H H∝V1/2 Reheating Inﬂation ρ∝V(φ) ln ln Radiation dominated H∝a-2 Matter dominated H∝a-3/2 Λ dominated H∝Λ1/2 ln a Axion production

Thursday, April 26, 12 Axion CDM - Standard cosmology

! 10 12 18 Θ " 10 i 0.0001

Θ " i 0.001 1016 Axion isocurvature PLANCK !9 fluctuations 10 Θ " i 0.01 " 14 " 10 PLANCK eV !

GeV Θ " i 0.1 ! ! 6 a

a 10 m f 12 ADMXI 10 Θ " ADMXII Tensor modes i 1 #a $# CARRACK CDM 10 T GH # %# Θ " " a CDM !3 10 i 3.14 f a 10

108 White dwarfs cooling time 104 106 108 1010 1012 1014

HI !GeV"

Thursday, April 26, 12 Axion CDM - Standard cosmology

! 10 12 18 Θ " 10 i 0.0001

Θ " i 0.001 1016 Axion isocurvature PLANCK !9 fluctuations 10 Θ " i 0.01 " 14 " 10 PLANCK eV !

GeV Θ " i 0.1 ! ! 6 a

a 10 m f 12 ADMXI 10 Θ " ADMXII Tensor modes i 1 #a $# CARRACK CDM 10 T GH # %# Θ " " a CDM !3 10 i 3.14 f a 10

108 White dwarfs cooling time 104 106 108 1010 1012 1014 PQ symmetry breaks before inﬂation ends PQ symmetry breaks after inﬂation ends HI !GeV"

Thursday, April 26, 12 Axion CDM - Standard cosmology

PQ symmetry breaks ! 10 12 after18 inﬂationΘ " ends 10 i 0.0001

• Average θi overΘ " Hubble& $volume , i 0.001 16 1 PLANCK ! • Anharmonicities10 are& important& Axion isocurvature 9 fluctuations 10 Θ " i 0.01 " $9:14 " 10 PLANCK ; $$7 eV !

GeV Θ " i 0.1 7/6 ! 2 ! e⇡ 6 a

a f(✓)= ln 2 2 10 ⇡ ✓ m f 12 ADMXI 10 Θ h" i ADMXII Tensor modes i 1 #a $# CARRACK CDM 10 T GH # %# Θ " θ/π " a CDM !3 10 i 3.14 10 f a 2 2 ✓i f(✓i) =(2.96) h 8 i White dwarfs cooling time String 10decay contribution is • 4 6 8 10 12 14 ~16% of 10vacuum realignment10 10 10 10 10 Axionduring oscillations inﬂation start PQ symmetry breaks after inﬂation ends HI !GeV"

Thursday, April 26, 12 Axion CDM - Standard cosmology

PQ symmetry breaks!12 before inﬂation10 ends 18 Θ "0.0001 10 i • Constrained by non-adiabatic Θ " i 0.001 ﬂuctuations

16 PLANCK ! 10 Axion• Single isocurvature value of θi throughout9 fluctuations 10 Θ " i 0.01 Hubble volume 2 " 2 2 HI 14 ✓i f(✓i) = ✓i + f " (✓i) 10 PLANCK h i " 2⇡fa # ✓ ◆ eV !

GeV Θ " ! " i 0.1 ma eV ! ! ! ! ! ! ! 6 ! !a 10 3 10 4 10 5 10 6 10 7 10 8 10 9 a 10 m f 12 ADMXI1.0 10 Θ " ADMXII StrongTensor modes CP i 1 # $# 0.8 a CDM AnharmonicitiesCARRACK problem? 10 T GH # %# Θ " " a CDM !3 10 i 3.14 0.6 a ! "!10 f Π a CDM # i Θ 0.4 8 White dwarfs cooling time! #! 10 a CDM 104 106 108 1010 0.21012 1014 PQ symmetry breaks before inﬂation ends Axion0.0 oscillations start HI !GeV" after109 end1010 of10 inﬂation11 1012 1013 1014 1015 1016 fa !GeV"

Thursday, April 26, 12 Axion CDM - Standard cosmology

! 10 12 18 Θ " 10 i 0.0001

Θ " i 0.001 1016 Axion isocurvature PLANCK !9 fluctuations 10 Θ " i 0.01 " 14 " 10 PLANCK eV !

GeV Θ " i 0.1 ! ! 6 a

a 10 m f 12 ADMXI 10 Θ " ADMXII Tensor modes i 1 #a $#CDM CARRACK ma = 85 µeV 10 T GH # %# Θ " " a CDM !3 10 i 3.14 f a 10

108 White dwarfs cooling time 104 106 108 1010 1012 1014

HI !GeV"

Thursday, April 26, 12 Low Temperature Reheating cosmology

Thursday, April 26, 12 Axion CDM - Low Temp. Reheating cosmology

18 ! 10 TRH 4MeV ! TRH 15MeV ! TRH 150MeV Standard 16 10 Axion Isocurvature 10!9 Fluctuations

" 14 10 " eV !

GeV !6 Tensor Modes

! 10 ADMX a a 12 m f 10

Π !2 H I 10 ! !3 10 f a 10

108 White Dwarfs Cooling Time 104 106 108 1010 1012 1014

HI !GeV"

Thursday, April 26, 12 Axion CDM - Low Temp. Reheating cosmology

18 ! 10 TRH 4MeV ! TRH 15MeV ! TRH 150MeV Standard 16 10 Axion Isocurvature 10!9 Fluctuations

" 14 10 " eV !

GeV !6 Tensor Modes

! 10 ADMX a a 12 m f 10

Π !2 H I 10 ! !3 10 f a 10

108 White Dwarfs Cooling Time 104 106 108 1010 1012 1014 PQ symmetry breaks before inﬂation ends PQ symmetry breaks after inﬂation ends HI !GeV"

Thursday, April 26, 12 Axion CDM - Low Temp. Reheating cosmology PQ symmetry breaks 18 ! 10 TRH 4MeV ! after inﬂationTRH ends15MeV ! TRH 150MeV Standard • As TRH 16decreases, fa must 10 Axion Isocurvature 10!9 increase and ma decrease Fluctuations

10!8 Tensor Modes lowest TRH 14 " 14 10 10 0.1 µeV " LTR fa " HI !2Π fa !6 lower TRH eV 10 !

GeV !6 Tensor Modes

! ADMX 10 12 a

" 10

ADMX" a 12 m f

10 eV ! GeV std ! fa !4 a a

10 m f Π 10 std 10 1 !2 T Big Bang Nucleosynthesis I ! H ! 10 RH T !3 10 10!2 f a 10 108 White Dwarfs Cooling Time 5 10 50 100 500 1000 8 White Dwarfs Cooling Time 10 TRH !MeV" 104 106 108 1010 1012 1014 PQ symmetryduring inﬂation breaks PQ symmetry breaks after inﬂation ends HI !GeV"

Thursday, April 26, 12 Axion CDM - Low Temp. Reheating cosmology PQ symmetry breaks 18 ! 10 TRH 4MeV ! TRH 15MeV before inﬂation ends ! TRH 150MeV Standard • As TRH decreases, constraints 16 10 Axionfrom Isocurvature non-adiabatic ﬂuctuations10!9 Fluctuations lower TRH become weaker

" 14 10 • And the required initial " misalignment angle θi is larger eV !

GeV m !eV" !6

a Tensor Modes

! 10 ADMX !1 !2 !3 !4 !5 !6 !7 !8 !9 a !10 a 12 10 10 10 10 10 10 10 10 10 10 m f 10 1.0

fa " HI !2Π Π lower TRH !20.8 H I 10 ! 10 f a !3 0.6 10 !a "!CDM Π # i Θ GUT 8 0.4 10 White Dwarfs Cooling Time scale 4 6 8 10 0.2 12 14 ! #! 10 10 10 10 10 White Dwarfs Cooling Time 10a CDM PQ symmetry breaks before inﬂation ends PQ0.0 symmetry breaks 7 8 9 10 11 12 13 14 15 16 17 HI !GeV" after10 end10 of10 inﬂation10 10 10 10 10 10 10 10 fa !GeV"

Thursday, April 26, 12 Kination cosmology

Thursday, April 26, 12 Axion CDM - Kination cosmology

10!12 18 ! Tkin 4MeV 10 ! Tkin 300MeV ! Tkin 700MeV Standard 16 10 Axion Isocurvature 10!9 Fluctuations " 1014 " eV ! GeV !6 ! a Tensor Modes 10

a ADMX m f 1012

Π !2 10 H I 10 ! !3 f a 10

108 White Dwarfs Cooling Time 104 106 108 1010 1012 1014

HI !GeV"

Thursday, April 26, 12 Axion CDM - Kination cosmology

10!12 18 ! Tkin 4MeV 10 ! Tkin 300MeV ! Tkin 700MeV Standard 16 10 Axion Isocurvature 10!9 Fluctuations " 1014 " eV ! GeV !6 ! a Tensor Modes 10

a ADMX m f 1012

Π !2 10 H I 10 ! !3 f a 10

108 White Dwarfs Cooling Time 104 106 108 1010 1012 1014 PQ symmetry breaks before inﬂation ends PQ symmetry breaks after inﬂation ends HI !GeV"

Thursday, April 26, 12 Axion CDM - Kination cosmology

PQ symmetry breaks 10!12 18 ! after inﬂationTkin ends4MeV 10 ! Tkin 300MeV ! Tkin 700MeV • As Tkin decreases, Standardfa must 16 decrease10 and ma increase Axion Isocurvature 10!9 Fluctuations

• String" decay contribution is 15 × 10vacuum14 realignment " eV ! GeV !6 ! a Tensor Modes 10 10!8 a ADMX m f 12 Tensor Modes 1014 10

fa " HI !2Π !6 10 Π 2 ADMX ! 12 10 H I " 10 10 " ! lower Tkin !3 a 10

eV f ! GeV std ! fa !4 a a

10 m f std 1 10 T 10 ! Big Bang Nucleosynthesis 8 White Dwarfs Cooling Time 15 meV kin

10 T kin fa 104 106 108 10!2 1010 1012 1014 108 PQ symmetryWhiteduring Dwarfs inﬂation Cooling breaks Time 5 10 50 100 500 1000 PQ symmetry breaks !GeV" after inﬂation ends Tkin !MeV" HI

Thursday, April 26, 12 Axion CDM - Kination cosmology

PQ symmetry breaks10!12 18 ! before inﬂation ends Tkin 4MeV 10 ! Tkin 300MeV ! Tkin 700MeV • As Tkin decreases, constraints Standard 16 from non-adiabatic ﬂuctuations 10 Axion Isocurvature 10!9 lower Tkin becomeFluctuations stronger "

• And the required initial " 1014 misalignment angle θi is smaller eV !

GeV eV ma ! " !6 ! a !1 !2 !3 !4 !5 Tensor Modes !610!7 !8 !9 !10

a ADMX 10 10 10 10 10 10 10 10 10 10 m f 1012 1.0 fa " HI !2Π 0.8 Π !2 10 H I 10 ! !3 f a 0.6 10 !a "!CDM Π # i Θ 8 White Dwarfs Cooling0.4 Time 10 lower Tkin 4 6 8 10 0.2 12 14

10 10 10 10 10 White Dwarfs Cooling Time 10 PQ symmetry breaks ! #! before inﬂation ends PQ0.0 symmetrya breaksCDM 7 8 9 10 11 12 13 14 15 16 17 HI !GeV" after10 end10 10of inﬂation10 10 10 10 10 10 10 10 fa !GeV"

Thursday, April 26, 12 Conclusions For axions to be 100% of cold dark matter..... • If the Peccei-Quinn symmetry breaks after inﬂation ends, the axion mass must be ma=83 µeV in standard cosmology

- much smaller ma in LTR cosmology 0.1 µeV

• If the Peccei-Quinn symmetry breaks before inﬂation ends, an initial misalignment angle θi can be chosen for any ma<15 meV

- larger allowed region and larger θi in LTR cosmology - smaller allowed region and smaller θi in kination cosmology

Thursday, April 26, 12