Listening for Invisible Axions with Gravitational Waves Ben A. - - PowerPoint PPT Presentation

Listening for Invisible Axions with Gravitational Waves

Ben A. Stefanek

Audible Axions

• C. Machado, W. Ratzinger, P

. Schwaller, BAS JHEP 01 (2019) 053 arXiv:1811.01950

The Audible Axion Model (4 parameters)

• We consider the following effec3ve field theory consis3ng of an

axion-like par3cle (ALP) and a massless dark photon

• We assume some explicit breaking of the global symmetry at the

scale , which generates a mass for the ALP

2

S = Z d4x √−g 1 2∂µφ ∂µφ − V (φ) − 1 4XµνXµν − α 4f φXµν e Xµν

• <latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit><latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit><latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit><latexit sha1_base64="AQaBXR2yDKoGtrmXTFD4PAe7Hpw=">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</latexit>

φ

Xµ

V (φ) = m2f 2  1 − cos ✓φ f ◆

φi = θf , φ0

i ≈ 0 ,

θ ∼ O(1)

Λ = p mf

Vacuum Misalignment

• Since the ALP has no reason to be near the minimum of the

poten3al when it 3lts, we generically expect ini3al condi3ons of the form

3

φi = θf , φ0

i ≈ 0 ,

θ ∼ O(1)

V (φ)

φ

φi = θf

ALP and Dark Photon Abundances

• The ALP begins to oscillate when the Hubble rate drops below the

ALP mass

• For simplicity, we assume the no ini3al abundance for the dark
• photon. However, it can be sourced as the ALP rolls.

4

V (φ)

φ

φi = θf

Ωosc

φ

= ρosc

φ

ρosc

tot

≈ m2θ2f 2/2 3M 2

P H2

• sc

≈ ✓ θf MP ◆2

ρosc

φ

∼ 1 2m2φ2

i

Hosc ∼ m ,

Dark Photon Production

• We assume that the ALP begins oscilla3ng in a post-infla3onary

radia3on-dominated FRW universe ( ), in which the ALP

• beys

and the dark photon obeys a wave equa3on sourced by

5

✓ @2 @⌧ 2 r2 ◆ ~ X = ↵0 f ~ r ⇥ ~ X , ~ r · ~ X = 0 .

00 + 2aH0 + a2 @V @ = ↵ f a2 ~ E · ~ B ,

φ0

Ωφ < Ωrad

Source for dark photons (while ALP rolls) Dark E and B, sink for ALP

Dark Photon Production: Quantum Picture

We quan3ze the dark photon field as which leads to the following equa3on for the mode func3ons As the ALP rolls, there exist momenta for which is nega3ve. The corresponding modes are tachyonic and grow exponen3ally:

6

ˆ Xi(x, τ) = X

λ=±

Z d3k (2π)3 vλ(k, τ) εi

λ(k) ˆ

aλ(k) eik·x+ h.c.

v±(k, τ) ∼ e|ω±|τ

ω2

±

Circular pols. ( )

v00

± + ω2 ±(τ)v± = 0 ,

ω2

±(τ) = k2 ⌥ k α

f φ0

k ⇥ ε± = ⌥ikε±

Dark Photon Spectrum and Parity Violation

Because the ini3al sign of determines which helicity becomes tachyonic first, the system violates parity and one helicity experiences exponen3ally more produc3on than the other. Tachyonic modes: Most tachyonic mode:

7

2 5 10 20 50 10-25 10-20 10-15 10-10 10-5 1 / ( ) Ω /

φ0

ω2

+(τ) = k2 − k α

f φ0

˜ k(τ)

0 < k < αφ0 f , k m . αθ ˜ k = αφ0 2f . αθ 2 m

Gravitational Waves

Dark photon modes in the range which were ini3ally in vacuum grow exponen3ally when the axion begins to oscillate These rapidly growing modes amplify quantum fluctua3ons of the dark photon into a 3me-varying, anisotropic classical energy distribu3on which sources gravita3onal waves

8

v±(k, τ) ∼ e|ω±|τ

ˆ Πij(k, τ) = Λkl

ij

a2 Z d3q (2π)3 h ˆ Ek(q, τ) ˆ El(k − q, τ) + ˆ Bk(q, τ) ˆ Bl(k − q, τ) i .

v±(k, τ ⌧ τosc) = 1 p 2k e−ikτ

h00

ij(k, τ) + k2hij(k, τ) =

2 M 2

P

Πij(k, τ) ,

Anisotropic stress Gravity Waves

ω2

± < 0

0 < k < θαm

The Big Picture

9

Cold Dark Matter

H . m, Particle Production

H > m

P . Agrawal, G. Marques-Tavares, W. Xue: arXiv:1708.05008

• The oscilla3ng ALP produces dark photons which source GW. ASer

par3cle produc3on, the remaining ALP abundance can be CDM.

Gravitational Wave Spectrum

10

10-2 0.1 1 10 100 10-13 10-12 10-11 10-10 10-9 10-8 10-7 / ( ) Ω

kpeak ≈ (αθ)2/3 m ,

ΩGW(kpeak) ≈ ✓ f MP ◆4 ✓θ2 α ◆ 4

3

.

Properties of the GW Spectrum: Chirality

The produced GW spectrum is highly polarized because the GW inherit the parity viola3on in the dark photon popula3on

11

2 5 10 20 50 10-25 10-20 10-15 10-10 10-5 1 / ( ) Ω / 10-2 0.1 1 10 100 10-13 10-12 10-11 10-10 10-9 10-8 10-7 / ( ) Ω

+ = + =

12

10-9 10-7 10-5 10-3 10-1 101 103 10-16 10-14 10-12 10-10 10-8 10-16 10-14 10-12 10-10 10-8

f0 HHzL h2WGW

LISA BBO DECIGO L I G O ET IPTA SKA QCD Axion 1 QCD Axion 2 A L P 1 ALP 2 ALP 3 ALP 4 A L P 5 ALP 6

Listening for Invisible Axions

Ω0

GW ≈ Ω0 γ

✓ f MP ◆4 ✓θ2 α ◆ 4

3

f0 ≈ m ✓T0 T∗ ◆ (αθ)2/3 = r m MP T0 (αθ)2/3 ,

13

10-9 10-7 10-5 10-3 10-1 101 103 10-16 10-14 10-12 10-10 10-8 10-16 10-14 10-12 10-10 10-8

f0 HHzL h2WGW

LISA BBO DECIGO LIGO ET IPTA SKA QCD Axion 1 QCD Axion 2 A L P 1 ALP 2 ALP 3 ALP 4 A L P 5 A L P 6

ALPs Detectable by Pulsar Timing Arrays:

f & 1017 GeV

m ∼ 10−13 - 10−10 eV

14

10-9 10-7 10-5 10-3 10-1 101 103 10-16 10-14 10-12 10-10 10-8 10-16 10-14 10-12 10-10 10-8

f0 HHzL h2WGW

LISA BBO DECIGO LIGO ET IPTA SKA QCD Axion 1 QCD Axion 2 A L P 1 ALP 2 ALP 3 ALP 4 A L P 5 A L P 6

ALPs Detectable by LISA:

m ∼ 0.01 − 1 eV

f & 1017 GeV

15

10-9 10-7 10-5 10-3 10-1 101 103 10-16 10-14 10-12 10-10 10-8 10-16 10-14 10-12 10-10 10-8

f0 HHzL h2WGW

LISA BBO DECIGO LIGO ET IPTA SKA QCD Axion 1 QCD Axion 2 A L P 1 ALP 2 ALP 3 ALP 4 A L P 5 A L P 6

Future High Frequency Detectors:

f & 1017 GeV

m ∼ 102-106 eV

Listening for Invisible Axions

• Because large decay constants correspond to larger GW amplitude, gravita3onal

wave detectors can probe Audible Axion models from the “boVom up”!

16

10-14 10-9 10-4 10 106 10-18 10-16 10-14 10-12 10-10 10-8 10-6

m [eV] 1/fa [GeV-1]

SKA LISA BBO DECIGO ET Astrophysics IAXO ADMX Q C D A x i

• n

Neff Cosmological BH Superradiance

Listening for Invisible Axions

• Because large decay constants correspond to larger GW amplitude, gravita3onal

wave detectors can probe Audible Axion models from the “boVom up”!

17

10-14 10-9 10-4 10 106 10-18 10-16 10-14 10-12 10-10 10-8 10-6

m [eV] 1/fa [GeV-1]

SKA LISA BBO DECIGO ET Astrophysics IAXO ADMX Q C D A x i

• n

Neff Cosmological BH Superradiance

Thank You!

Backup Slides

Properties of the GW Spectrum: Param Scaling

The energy in GW is limited by the energy in the axion field Rough es3mate for the peak of the GW spectrum Es3ma3ng at the 3me of emission leads to

19

Ωosc

φ

= ρosc

φ

ρosc

tot

≈ m2θ2f 2/2 3M 2

P H2

• sc

≈ ✓ θf MP ◆2 . kpeak ≈ ˜ k(τ∗) , ΩGW(kpeak) = Ω2

φ

✓a∗H∗ kpeak ◆2 .

ΩGW(kpeak) ≈ ✓ f MP ◆4 ✓θ2 α ◆ 4

3

.

kpeak ≈ (αθ)2/3 m ,

Large decay constant required ALP mass sets peak frequency

1 2 5 10 20 50 10-8 10-4 1 104 108 1012 / ρ (+) / ρ ΔNeff = 0.3 ΔNeff = 0.03

ρϕa3 ρXa4

Quick Recap

20

Cold Dark Matter

H . m, Particle Production

H > m

0.1 1 10 100 10-12 10-11 10-10 10-9 10-8 10-7 / ( ) Ω

a /aosc = 6.7 a /aosc = 10.3 a /aosc = 66

P . Agrawal, G. Marques-Tavares, W. Xue: arXiv:1708.05008

Energy Density and Gauge Power Spectrum

1 2 5 10 20 50 10-8 10-4 1 104 108 1012 / ρ (+) / ρ ΔNeff = 0.3 ΔNeff = 0.03

ρϕa3 ρXa4

1 2 5 10 20 50 10-43 10-33 10-23 10-13 10-3 / ( ) Ω /

a /aosc = 6.7 a /aosc = 10.3 a /aosc = 23 a /aosc = 66

More on GW Spectrum

ˆ hij(k, τ) = 2 aM 2

P

Z τ

τosc

dτ 0a(τ 0)G(k, τ, τ 0)ˆ Πij(k, τ 0) dρGW d log k = M 2

P k3

8π2a2 Ph0(k, τ)

h0|ˆ h0

ij(k, τ)ˆ

h

0⇤

ij(k0, τ)|0i = (2π)3Ph0(k, τ)δ(k k0)

dρGW d log k = k3 4π2a4M 2

P

Z τ

τosc

dτ 0dτ 00a(τ 0)a(τ 00) cos [k(τ 0 − τ 00)] Π2(k, τ 0, τ 00)

h0|ˆ Πij(k, τ)ˆ Π⇤

ij(k0, τ 0)|0i = (2π)3Π2(k, τ, τ 0)δ(k k0)