dark matter axion condensates: stability and ...

DARK MATTER AXION CONDENSATES: STABILITY AND ASTROPHYSICAL PROPERTIES (arXiv: 1710.04729; 1804.07255; 1805.00430 )

Enrico D. Schiappacasse (with Mark P. Hertzberg)

Tufts University

12th International Conference Identification of Dark Matter 2018 Brown University

Tufts Institute of Cosmology Tufts University

2

INTRODUCTION

• •

A popular dark matter candidate is the QCD axion: PG-boson associated with SSB of 𝑈 1 𝑃𝑄 , which was introduced as a possible solution of the strong CP problem (Preskill et al., 1983; Peccei and Quin, 1977; Weinberg, 1978) We are interested in small scale axion substructure:

(Sikivie and Yang, 2009)

On small scales axions can gravitationally thermalize leading to a type of BEC (Guth, Hertzberg, and Prescod-Weinstein, 2015)

Condensate of short range order driven by attractive interactions : gravity + self interactions 𝜆𝜙 4 (Asztalos et al., 2010; Hoskins, 2011)

•

Axions couple to photons via the operator: Δℒ~𝑔𝑎𝛾 𝜙 𝑬 ∙ 𝑩

•

We explore the possibility of parametric resonance of photons in the context of dark matter axion clumps.

This coupling is exploited to try to detect axions in ground based experiments, such as the ADMX

ENRICO D. SCHIAPPACASSE – IDM 2018 BROWN UNIVERSITY – DARK MATTER AXION CONDENSATES

3

AXION FIELD THEORY

• •

Axions are described in field theory by a real scalar field 𝜙 𝑥, 𝑡 with a small potential 𝑉(𝜙) coming from nonperturbative QCD effects. 1 2

2 2 Expanding around the CP preserving vacuum: 𝑉 𝜙 = 𝑚𝜙 𝜙 +

𝜆 4 𝜙 4!

+⋯

We are interested in the non-relativistic regime (field configurations where 𝜙 is small)

•

For the standard QCD axion We shall often use 𝑚𝜙 ~10−5 eV and 𝑓𝑎 = 6 × 1011 GeV

•

2 𝑚𝜙

=

2 𝑚𝜙

𝑚𝑢,𝑑,𝜋 , 𝑓𝜋 , 𝑓𝑎 and 𝜆 = −𝛾

2 𝑚𝜙

𝑓𝑎2

𝟎 ∶ Exponentially growing solutions (First instability band) For ℜ(𝜇𝒌 ) = 𝟎 : Instability band edges 𝑘𝒍/𝒓𝒆𝒅𝒈𝒆 =

𝑘 ∗ = 𝜔0 /2

𝝎𝟒𝟎 𝒈𝟐𝒂𝜸 𝝎𝟐𝟎 𝝓𝟐𝟎 𝒈𝒂𝜸 𝝎𝟎 𝝓𝟎 + ∓ 𝟒 𝟏𝟔 𝟒

1 + 𝒈𝟐𝒂𝜸 𝝓𝟐𝟎 /2 ≈ 𝑚𝜙 /2

Maximum Floquet Exponent

Center of the band

∗ 𝜇𝐻 ≈

𝒈𝒂𝜸 𝒎𝝓 𝝓𝟎 4

(Hertzberg and Schiappacasse, 1805.00430 )


SPHERICALLY SYMMETRIC CLUMP CONDENSATES Clump Profile

• • •

9

Gravity will inevitably cause a homogeneous condensate to fragment into an inhomogeneous field configuration: locally this leads to the formation of BEC clumps. This axion-gravity-self-interacting system shows a stable branch of solutions in the non-relativistic regime, where gravity is the dominant interaction. The true BEC ground state is guaranteed to be spherically symmetric: 𝜓 𝑟, 𝑡 = Ψ(𝑟)𝑒 −𝑖𝜇𝑡

• axion-gravity-self-interacting system SECH ANSATZ

Ψ 𝑟 = Ψ0 sech(𝑟/𝑅) with Ψ0 =

Ψ(𝑟) describes the radial profile 𝜇 describes the correction to the frequency

3𝑁 𝜋3 𝑅3

The corresponding axion field 𝜙 in this non-relativistic regime is

2 2 𝐺𝑚𝜙 𝑁 𝑁 𝜆𝑁 2 H 𝑅 =𝑎 −𝑏 +c 2 3 𝑚𝜙 𝑅 2 𝑅 𝑚𝜙 𝑅

𝑁𝑚𝑎𝑥 ≈

10.12 𝜆

𝛿

; 𝑅𝑚𝑖𝑛 ≈

0.13 𝑚𝜙 𝛿

with 𝛿 =

2 𝐺𝑚𝜙

𝜆

=

𝐺𝑓𝑎2 𝛾

𝜙 𝑟, 𝑡 =

2 Ψ 𝑚𝜙

with 𝜙0 =

R is the effective radius of the solution (variational parameter) (Schiappacasse and Hertzberg, 1710.04729 )

𝑟 cos(𝜔0 𝑡) 2 Ψ 𝑚𝜙 0

(𝜔0 = 𝑚𝜙 + 𝜇 ≈ 𝑚𝜙 )


SPHERICALLY SYMMETRIC CLUMP CONDENSATES Vector Spherical Decomposition

• •

10

Since 𝜙 = 𝜙(𝑟, 𝑡), the usual 3-dimensional Fourier transform of the equation of motion for the vector potential is not the best way to proceed. We prefer performing a vector spherical harmonic decomposition of 𝑨𝒌 :

𝑑3 𝑘 𝑨 𝒙, 𝑡 = න ෍ 𝑣𝑙𝑚 𝑘, 𝑡 𝑴𝑙𝑚 𝑘, 𝒙 − 𝑤𝑙𝑚 𝑘, 𝑡 𝑵𝑙𝑚 (𝑘, 𝒙) (2𝜋)3

•

𝑙𝑚

Again neglecting gradients of the axion field,

Here 𝑴𝑙𝑚 , 𝑵𝑙𝑚 are vector spherical harmonics, 𝑖𝑗 (𝑘𝑟) 𝑖𝑚 𝜕𝑌 where 𝑴𝑙𝑚 = 𝑙 𝑌𝑙𝑚 𝜃෠ − 𝑙𝑚 𝜑ො and 𝑙(𝑙+1) sin 𝜃

𝜕𝜃

∇ × 𝑴𝑙𝑚 = −𝑖𝑘𝑵𝑙𝑚 , ∇ × 𝑵𝑙𝑚 = 𝑖𝑘𝑴𝑙𝑚

𝑨ሷ − ∇2 𝑨 + 𝑔𝑎𝛾 ∇ × 𝜕𝑡 𝜙 𝑨 = 0

•

𝑑3 𝑘 න ෍ 𝑣ሷ 𝑙𝑚 + 𝑘 2 𝑣𝑙𝑚 − 𝑖𝑘𝑔𝑎𝛾 𝜕𝑡 𝜙𝑤𝑙𝑚 𝑴𝑙𝑚 𝑘, 𝒙 − 𝑤ሷ 𝑙𝑚 + 𝑘 2 𝑤𝑙𝑚 + 𝑖𝑘𝑔𝑎𝛾 𝜕𝑡 𝜙𝑣𝑙𝑚 𝑵𝑙𝑚 (𝑘, 𝒙) = 0 3 (2𝜋) 𝑙𝑚

lf-interacting system

It can be solved numerically, but for any arbitrary sum over {l,m} is quite complicated



• •

•

11

We focus on the 𝑙 = 1, 𝑚 = 0 channel for simplicity and leave a complete analysis for future work. መ 𝜑ො . Since the Coulomb Gauge reduces the system We write out the individual vector components 𝑟,Ƹ 𝜃, to only two independents equations ∇ ∙ 𝐴መ = 0 , we focus only on 𝑟,Ƹ 𝜑ො components.

For the radial component, use 𝜙 = Φ 𝑟 cos(𝜔0 𝑡) and orthogonality properties of Spherical Bessel functions to obtain

𝑤ሷ 10

𝑘′, 𝑡

+

2 𝑘 ′ 𝑤10

𝑘′, 𝑡

2𝑖 − 𝑔𝑎𝛾 𝜔0 𝑘 ′ sin(𝜔0 𝑡) න 𝑑𝑘 𝑘 2 𝑣10 (𝑘, 𝑡) න 𝑑𝑟 𝑟 2 Φ 𝑟 𝑗1 𝑘𝑟 𝑗1 𝑘 ′ 𝑟 = 0 𝜋

The spherically symmetry of the axion field means that we can ෩ 1𝑑 (𝑘) represent Φ 𝑟 by a 1-d (real) Fourier transform Φ 𝑑 𝑘෨ ෨ Φ ෨ ෩ 1𝑑 (𝑘) Φ 𝑟 =න cos(𝑘𝑟) 2𝜋 The 1d Fourier transform is dominated by 𝑘෨ ≈ 0. For example for sech ansatz ෪ Φ1𝑑 𝑘෨ =

෨𝑅 3𝑁 𝜋𝑘 sech 𝜋𝑅 2

෨ , we have 𝑘~1/𝑅 ≪ 𝑚𝜙

≈

𝑘 2 +𝑘′2 8𝑘 2 𝑘′2

෩ 1𝑑 𝑘 − 𝑘 ′ − Φ ෩ 1𝑑 𝑘 + 𝑘 ′ Φ 𝑚𝜙

The resonance occurs when 𝑘 ≈ 𝑘 ′ ≈ → the second 2 term is exponential suppressed for 𝑅 ≫ 1/𝑚𝜙



•

A similar procedure can be applied for the other angular component equation, to obtain 𝑤ሷ 10 𝑘, 𝑡 +

𝑘 2 𝑤10

𝑣ሷ10 𝑘, 𝑡 + 𝑘 2 𝑣10

• •

12

𝑑𝑘′ ෩ 1𝑑 𝑘 − 𝑘 ′ =0 𝑘, 𝑡 − 𝑖𝑔𝑎𝛾 𝜔0 𝑘 sin(𝜔0 𝑡) න 𝑣10 (𝑘′, 𝑡)Φ 2𝜋 𝑑𝑘′ ෩ 1𝑑 𝑘 − 𝑘 ′ =0 𝑘, 𝑡 + 𝑖𝑔𝑎𝛾 𝜔0 𝑘 sin(𝜔0 𝑡) න 𝑤10 (𝑘′, 𝑡)Φ 2𝜋

A self-consistent resonant solution is just given by 𝑤10 𝑘, 𝑡 = ±𝑖𝑣10 (𝑘, 𝑡) , which reduces the system to a single scalar differential equation We compute the resonance structure numerically using Floquet theory : In the homogeneous case, there always exists a non-zero maximum Floquet ∗ exponent 𝜇𝐻 ≈ 𝑔𝑎𝛾 𝑚𝜙 𝜙0 /4

interacting system

(1) We determine the maximum Floquet exponent 𝜇∗ (2) We use various choices of 𝑔𝑎𝛾 and parameters of axion clump (𝑅 and 𝑁) (3) We operate in the sech approximation on the stable branch


SPHERICALLY SYMMETRIC CLUMP CONDENSATES Numerical Results

13 (Hertzberg and Schiappacasse, 1805.00430 )

The real part of 𝜇 ∗ becomes zero below a critical 𝑔𝑎𝛾 or critical 𝑁. For the QCD axions, when 𝑁 = 𝑁𝑚𝑎𝑥 and 𝛾 = 0.3, parametric resonance

• system

happens for 𝑔𝑎𝛾 > 𝑔𝑎𝛾,𝑚𝑖𝑛 ≈

𝛽𝑐 𝑓𝑎

with

𝛽𝑐 ≈ 0.3 In conventional QCD axion models, 𝑔𝑎𝛾 =

𝑂(10−2 ) 𝑓𝑎

. So, the resonance

could be possible in unconventional axions models or for couplings to hidden sector photons The maximum real part of Floquet exponent 𝜇 ∗ , describing parametric resonance of photons from a spherically symmetric clump 𝛾 ෩ = 𝑁/(|𝜆| 𝛿). condensate, as a function of axion-photon coupling 𝑔𝑎𝛾 . (Left) We plot 𝜇 ∗ in units of 𝑚𝜙 𝛿 , 𝑔𝑎𝛾 in units of , and 𝑁

(Daido, Takahashi and N. Yokozaki 2018)

𝑓𝑎

∗

(Right) We plot 𝜇 in units of 𝑚𝜙 𝛿 and 𝑁 in units of 𝜆

∗ Resonance Condition : 𝜇𝐻 ≈

𝑔𝑎𝛾 𝑚𝜙 𝜙0 4

Hertberg (2010); Kawasaki and Yamada (2014)

1

> 𝜇𝑒𝑠𝑐 ≈ 2𝑅

−1 −1/2

𝛿

. Here 𝑔෤𝑎𝛾 = 𝑔𝑎𝛾 𝑓𝑎 / 𝛾.

Furthermore, excellent approximation to the growth rate from a ∗ 𝜇𝐻 − 𝜇𝑒𝑠𝑐 , localized clump: 𝜇 ≈ ቊ 0, ∗

∗ 𝜇𝐻 > 𝜇𝑒𝑠𝑐 ∗ 𝜇𝐻 < 𝜇𝑒𝑠𝑐


CLUMP CONDENSATES WITH ANGULAR MOMENTUM Non-Spherical Clump Profile

•

14

Clump condensates with non-zero angular momentum have a larger 𝑁𝑚𝑎𝑥 (and field amplitude), which will help the resonance phenomenon. ∞

The angular momentum is 𝐋 = 0,0, 𝑁m with 𝑁 = 4𝜋 ‫׬‬0 𝑑𝑟 𝑟 2 Ψ(𝑟)2

• • •

We take the field profile to be 𝜓 𝑥, 𝑡 = 4𝜋Ψ(𝑟)𝑌lm (𝜃, 𝜑)𝑒 −𝑖𝜇𝑡 We look for states which minimize the energy at fixed particle number and fixed angular momentum As usual we make an ansatz for the radial profile Ψ(𝑟): For non-zero l, the structure for small r behavior drastically changes in comparison to the l=0 case 1 2

We need Ψ 𝑟 = Ψ𝛼 𝑟 l − Ψ𝛽 𝑟 l+2 + ⋯ (near region)

𝜇𝑒𝑓𝑓 Ψ ≈ −

1 2𝑚𝜙

2 𝑟

Ψ ′′ + Ψ ′ +

It includes the gravitational term

l(l+1) Ψ 2𝑚𝜙 𝑟 2

(near region)

These terms blow up when 𝑟 ⟶ 0

The Hamiltonian is a generalization of the previous one for the 𝑙 = 0 case: constant coefficients (𝑎, 𝑏, 𝑐) become {l, m}-dependent

MODIFIED GAUSSIAN ANSATZ Ψ(𝑟) =

𝑟 l −𝑟 2 /(2𝑅 2 ) 𝑒 1 2𝜋 l+ !𝑅 3 𝑅 𝑁

2

2 2 𝐺𝑚𝜙 𝑁 𝑁 𝜆𝑁 2 H 𝑅 = 𝑎lm − 𝑏lm + 𝑐lm 2 3 𝑚𝜙 𝑅 2 𝑅 𝑚𝜙 𝑅



෩𝑅 = Ψ𝑅 𝑅 3 /𝑁 versus radius 𝑟෤ = 𝑟/𝑅 in the Field Ψ modified Gaussian ansatz for different values of spherical harmonic number 𝑙.

We plot 𝑅90 (0.9𝑁 = 4𝜋 ‫׬‬0𝑅90 𝑑𝑟 ′ 𝑟 ′ 2Ψ(𝑟′)2 ), where ෩ = 𝜆 𝛿 1/2 𝑁. 𝑅෨90 = 𝑚𝜙 𝛿 1/2 𝑅90 and 𝑁

(Hertzberg and Schiappacasse, 1804.07255 ) Energy of clump solution versus number with non-zero angular momentum parameter 𝑚 = 2 for different values of 𝑙. At a fixed number N and angular momentum 𝐿𝑧 = 𝑁𝑚, this illustrates that the configurations that minimizes the energy has spherical harmonic number 𝑙 = 𝑚 (whenever the solution 3 2 ෩ = 𝜆 2 / 𝑚𝜙 ෩ = 𝑚𝜙 𝐺 |𝜆|. exists). Here 𝐻 𝐺 𝐻 and 𝑁

At high 𝑙 = 𝑚 , 𝑁𝑚𝑎𝑥

𝑙3/2 0.141 𝑙1/2 ≈ ; 𝑅𝑚𝑖𝑛 ≈ 𝜆 𝛿 (ln 𝑙)1/4 𝑚𝜙 𝛿 (ln 𝑙)1/4 10.52


15

CLUMP CONDENSATES WITH ANGULAR MOMENTUM Approximate Treatment of Clump Resonance

• •

∗ For an approximate treatment of clump resonance, we use the condition for resonance 𝜇𝐻 > 𝜇𝑒𝑠𝑐 ,which was established for spherically symmetric clumps. lllll,

∗ Since 𝜇𝐻 ≈ 𝑔𝑎𝛾 𝑚𝜙 𝜙0 /4, we need to determine the maximum field amplitude:

𝜙0 =

2 𝑚𝜙

Ψ0

4𝜋 𝑌𝑙𝑚

4𝜋 𝑌𝑙𝑚 Ψ0 = ∗ 𝜇𝐻

16

𝑁𝑚𝑎𝑥 𝑓 𝑅𝑚𝑖𝑛 3 𝑙

≈ 5.8

with 𝑓𝑙 ≈

1 (2𝜋)3/4 𝑙

𝑙(ln 𝑙)1/4 𝑔෤𝑎𝛾 𝑚𝜙 1 4

0

≈

2𝑙1/4 𝜋1/4

𝜇𝑒𝑠𝑐 ≈ 3.5 ln 𝑙 𝑚𝜙 𝛿

The radial profile is now peaked around 𝑅𝑝 = 𝑙𝑅, but the full width in the radial direction is still 𝜔𝑟 ~2𝑅:

Ψ(𝑟) ≈ 0

•

𝑁 (2𝜋)3/2 𝑙𝑅

3𝑒

− 𝑟−𝑅𝑝

2

/𝑅2

for high l

The angular dependence is non-trivial. For 𝑙 = 𝑚 , the real field 𝜙 is

for high 𝑚 = 𝑙

for high 𝑚 = 𝑙

𝛿

•

𝜙 𝑟, 𝜃, 𝜑, 𝑡 =

2 𝑚𝜙

4𝜋 𝑌𝑙𝑚 0 Ψ(𝑟) − sin 𝜃 𝑙 cos(𝜔0 𝑡 ± 𝑙𝜑),

and the full width in the polar direction is 𝜔𝜃 ~∆𝜃𝑅𝑝 /𝜋~2𝑅 where ∆𝜃 = 2cos

−1

(𝑒

1 𝑙

− ln 2

). Then, we take as 𝜇𝑒𝑠𝑐 ~1/2𝑅

(for high 𝑚 = 𝑙 and 𝑁 = 𝑁𝑚𝑎𝑥 and 𝑔෤𝑎𝛾 =𝑔𝑎𝛾 𝑓𝑎 / 𝛾)



2 17

∗ 𝜇∗ ≈ 𝑀𝑎𝑥(𝜇𝐻 − 𝜇𝑒𝑠𝑐 , 0)

The minimum axion-photon coupling 𝑔𝑎𝛾 that is necessary in order to have resonance from a clump condensate as a function of its angular momentum 𝑚 = 𝑙. We plot in units of

𝛾 . 𝑓𝑎

We take

𝑁 = 𝑁𝑚𝑎𝑥 .

(Hertzberg and Schiappacasse, 1805.00430 ) By taking 𝛾 = 0.3 and 𝑁 = 𝑁𝑚𝑎𝑥 , for high angular momentum the minimum 𝛽 axion-photon coupling is just 𝑔𝑎𝛾 > 𝑔𝑎𝛾,𝑚𝑖𝑛 ≈ 𝑐 with 𝛽𝑐 = 0.3 The maximum real part of Floquet exponent 𝜇 ∗ , describing parametric resonance of photons from a clump condensate as a function of its angular momentum 𝑚 = 𝑙. We plot 𝜇 ∗ in units of 𝑚𝜙 𝛿. This is for attractive self-interactions with 𝑁 = 𝑁𝑚𝑎𝑥 . Here 𝑔෤𝑎𝛾 =

𝑔𝑎𝛾 𝑓𝑎 𝛾

.

𝑓𝑎 l

For the QCD axion 𝑔𝑎𝛾 = 𝑂(10−2 )/𝑓𝑎 , we need rather large angular momentum of 𝑚 = 𝑙 ≳ 𝑂(103 )


18

ASTROPHYSICAL CONSEQUENCES

•

• •

Suppose 𝑔𝑎𝛾 > 𝑔𝑎𝛾,𝑚𝑖𝑛 . The photon occupancy number will increase from 0 to a large value, then the final output is a essentially classical electromagnetic waves. After clumps formation, clumps with sufficiently large mass will undergo parametric resonance into photons. If 𝑁 > 𝑁𝑐 , the clump will radiate into photons, losing mass until 𝑀 → 𝑀𝑐 = 𝑁𝑐 𝑚𝜙 : PILE-UP AT A UNIQUE VALUE OF CLUMPS.

Clump radius R as a function of clump number 𝑁 for spherically symmetric clumps with attractive self-interactions. We have taken the axion-photon coupling to be 𝑔෤𝑎𝛾 =

𝑔𝑎𝛾 𝑓𝑎 𝛾

= 2 here. For any clumps on

the stable branch with number 𝑁 > 𝑁𝑐 they will resonantly produce photons, lose mass, and pile-up at the critical value 𝑁𝑐 ≈ 3.7/( 𝜆 𝛿).



OUTLOOK

•

• • •

In this work we have explored a possible novel consequence of the axion model, in which gravitationally axion bound clumps can form and undergo parametric resonance into electromagnetic radiation.

For conventional values of axion-photon coupling, BEC (ground state) of axion dark matter can not undergo parametric resonance. For BEC axion dark matter with sufficiently large angular momentum, atypically large axionphoton coupling 𝑔𝑎𝛾 ≳ 1Τ𝑓𝑎 , and for couplings to hidden sector, parametric resonance can occur. It would be interesting to further explore possible theoretical realizations of these more general possibilities as well as to explore possible hints of the idea of a clump mass pile-up.


19

REFERENCES • • • • • • • • • • • • • •

E. D. Schiappacasse and M. P. Hertzberg, “Analysis of Dark Matter Axion Clumps with Spherical Symmetry,” JCAP 1801, 037 (2018). M. P. Hertzberg and E. D. Schiappacasse, “Scalar Dark Matter Clumps with Angular Momentum,” arXiv:1804.07255 [hep-ph] (2018). M. P. Hertzberg and E. D. Schiappacasse, “Dark Matter Axion Clump Resonance of Photons,” arXiv:1805.00430 [hep-ph] (2018). J. Preskill, M. B. Wise and F. Wilczek, “Cosmology of the Invisible Axion,” Phys. Lett. B 120, 127 (1983). R. D. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons,” Phys. Rev. Lett. 40, 279 (1978). S. Weinberg, “A New Light Boson?,” Phys. Rev. Lett. 40, 223 (1978). J. E. Kim, ‘’Weak Interaction Singlet and Strong CP Invariance,’’ Phys. Rev. Lett. 43, 103 (1979). A. R. Zhitnitsky, “On Possible Suppression of the Axion Hadron Interactions. (In Russian),” Sov. J. Nucl. Phys. 31, 260 (1980). A. H. Guth, M. P. Hertzberg and C. Prescod-Weinstein, “Do Dark Matter Axions Form a Condensate with Long-Range Correlation?,” Phys. Rev. D 92, 103513 (2015).

G. Grilli di Cortana, E. Hardy, J. Pardo Vega and G. Villadoro, “The QCD axion, precisely,” JHEP 1601, 034 (2016). N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1947), Chap. 6. R. Daido, F. Takahashi and N. Yokozaki, “Enhanced axionphoton coupling in GUT with hidden photon,” Phys. Lett. B 780, 538 (2018). M. P. Hertzberg, “Quantum Radiation of Oscillons,” Phys. Rev. D 82, 045022 (2010). M. Kawasaki and M. Yamada, “Decay rates of Gaussian-type I-balls and Bose-enhancement effects in 3+1 dimensions,” JCAP 1402, 001 (2014).


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FURTHER DISCUSSIONS Physical Parameters for Axions

•

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We compute the maximum number of particles, the maximum mass, and the minimum clump size for axion clumps as follows: 𝑁𝑚𝑎𝑥 ~1.5 × 1060 Here ෡𝑓𝑎 ≡

𝑓𝑎 ,𝑚 ෝ𝜙 6 ×1011 GeV

≡

𝑎ො 𝑙 Ƹ 𝑏෠ 𝑙𝑚 𝑐𝑙𝑚 𝑚𝜙 10−5 eV

−2 መ −1/2 𝑚 ෝ𝜙 𝑓𝑎 𝛾ො ;

𝑀𝑚𝑎𝑥 ~2.5 × 1019 kg

, 𝛾ො ≡ 𝛾/0.3,

and the coefficients 𝑎ො𝑙 , 𝑏෠𝑙𝑚 , 𝑐𝑙𝑚 Ƹ are normalized to their zero angular momentum value.

𝑅90,𝑚𝑖𝑛 ~70 km

𝑅෨ 90 𝑅90

𝑎ො 𝑙 Ƹ 𝑏෠ 𝑙𝑚 𝑐𝑙𝑚

Ƹ 𝑐𝑙𝑚 𝑏෠ 𝑙𝑚

−1 መ −1/2 𝑚 ෝ𝜙 𝑓𝑎 𝛾ො

−1 መ −1 1/2 𝑚 ෝ𝜙 𝑓𝑎 𝛾ො

𝑎ො𝑙 ~𝑙/2, 𝑏෠𝑙𝑚 ~ (ln l)Τl , 𝑐𝑙𝑚 Ƹ ~ 1Τ 2l for high 𝑙 = |𝑚|

•

Note that the number of axions within a correlation length in the scenario in which the PQ symmetry is broken (Guth, Hertzberg, and Prescod-Weinstein, 2015) after inflation is given by system

𝑁𝜉 ~

3 𝑇𝑒𝑞 𝑀𝑝𝑙 3 𝑇𝑄𝐶𝐷 𝑚𝜙

−1 ~1061 𝑚 ෝ𝜙

For 𝑙 = 𝑚 ≳ 5, we have 𝑁𝑚𝑎𝑥 ≳ 1061


FURTHER DISCUSSIONS Effective Photon Mass

•

In the not-quite-empty space of the interstellar medium, photons acquire an effective mass equal to the plasma frequency as 𝜔𝑝2 =

• • •

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4𝜋𝛼𝑛𝑒 𝑚𝑒

=

𝑛𝑒 (6.4 0.03cm−3

× 10−12 eV)2

Considering the spatial distribution of 𝑛𝑒 and the fact that axion clump condensates are moving in the galactic halo: 𝜔𝑝 (𝑡) ≈ 𝜔𝑝 𝑓(𝑡), where 𝑓(𝑡) is a non-periodic time dependent function of orden 1. The modified equation for each polarization of the vector potential for the homogeneous case is 𝑨ሷ 𝑇𝑘 + [𝑘 2 +𝜔𝑝2 (𝑡) − 𝑔𝑎𝛾 𝜔0 𝑘𝜙0 sin(𝜔0 𝑡)]𝑨𝑻𝒌 = 0 Taking 𝑘 ≈ (𝑚𝜙 /2) and using as reference the amplitude 𝜙0 evaluated for the case of a sech ansatz, we have 2 𝜔𝑝

(𝑔𝑎𝛾 𝜔0 𝑘𝜙0 )

~

10−23 𝛽 10−2

10

−4 ~10 −19

So, we expect that the effect of the effective photon mass in the resonance should be negligible


FURTHER DISCUSSIONS Electromagnetic Emission in the Sky

•

•

•

We can imagine a scenario in which the process of resonance is still occurring. Consider a pair of clump condensates, each with number 𝑁1 and 𝑁2 (𝑁1 < 𝑁𝑐 and 𝑁2 < 𝑁𝑐 ).

Suppose that these clumps merge together in the late universe. If 𝑁𝑇𝑜𝑡𝑎𝑙 = 𝑁1 + 𝑁2 > 𝑁𝑐 , then the resonance will suddenly begin to occur, driving the total towards 𝑁𝑇𝑜𝑡𝑎𝑙 → 𝑁𝑐 . So, we expect a sudden emission of electromagnetic radiation in the galaxy:

𝜆𝐸𝑀 ≈ 2𝜋ൗ𝑘 ∗ ≈ 4𝜋ൗ𝑚𝜙

•

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𝜆𝐸𝑀 ~ 10−1 m

The typical mass of a clump is the order of ~10−11 𝑀⊙ , which is comparable to the moon’s mass. If the merger took place, an amount of energy comparable to 𝑀𝑐 2 equivalent to Moon’s mass would be emitted to the galaxy. ENRICO D. SCHIAPPACASSE – IDM 2018 BROWN UNIVERSITY – DARK MATTER AXION CONDENSATES

FURTHER DISCUSSIONS Estimate Growth time-scale

• • •

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Suppose 𝑔𝑎𝛾 > 𝑔𝑎𝛾,𝑚𝑖𝑛 . This will lead to an exponential growth in the electromagnetic field. The photon occupancy number will increase from 0 to a very large value, then the final output is essentially classical electromagnetic waves. ∗ We can estimate a lower bound on the time-scale for this growth by taking 𝜇∗ ~𝜇𝐻 . Let us consider the true BEC ground states. Since the condition for resonance is given by 𝑔𝑎𝛾 𝑓𝑎 > 0.3, we have

𝜇∗ ~15𝑔𝑎𝛾 𝑓𝑎 𝑚𝜙 𝛿 ≳ 5𝑚𝜙 𝛿 For typical values of QCD axion, we have 𝜏 = 1/𝜇 ∗ ≲ 𝑂(10−4 sec)


FURTHER DISCUSSIONS Repulsive Self-Interactions

•

For repulsive interactions we have only 𝜆 > 0 there is only 1 branch of solution. This branch is stable and there is no a maximum number of particles.


The maximum real part of Floquet exponent 𝜇 ∗ , describing parametric resonance of photons from a spherically symmetric clump condensate, as a function of axion-photon coupling 𝑔𝑎𝛾 . (Left) We plot 𝜇 ∗ in units of 𝑚𝜙 𝛿 , 𝑔𝑎𝛾 in units of

𝛾 , 𝑓𝑎

and

෩ = 𝑁/(|𝜆| 𝛿). (Right) We plot 𝜇 ∗ in units of 𝑁 𝑚𝜙 𝛿 and 𝑁 in units of 𝜆 −1 𝛿 −1/2 . Here 𝑔෤𝑎𝛾 = 𝑔𝑎𝛾 𝑓𝑎 / 𝛾.


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