Abhishek Banerjee - indico.ph.tum.de · Based On: With H Kim, G. Perez (1810.01889) + D. Budkar, J. Eby (1902.08212) + O. Matsedonskii (1912.04295) + Some unpublished speculations

Based On: With H Kim, G. Perez (1810.01889)

+ D. Budkar, J. Eby (1902.08212)

+ O. Matsedonskii (1912.04295)

+ Some unpublished speculations

RELAXION & HOW TO DETECT THEM!

Abhishek Banerjee Weizmann Institute of Science

Probing BSM physics at different scalesBerlin, Germany, 2020

RELAXION - 101

➤ Proposed to solve the hierarchy problem

Slide by Hyungjin Kim

Relaxion Dark Matter (DM)

V (�)

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Basic idea is similar to axion DM (but avoiding misalignment problem):

‣ After reheating the wiggles disappear (symmetry restoration):

V (�)

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Relaxion DM

V (�)

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‣ After reheating the wiggles disappear: and the relaxion rolls a bit


Relaxion DM

V (�)

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Relaxion DM

‣ After reheating the wiggles disappear: and the relaxion rolls a bit


V (�)

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‣ Universe cools down and the wiggles re-appear.


Relaxion DM

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‣ Now the relaxion not at the minimum and start to oscillate = DM.

Relaxion DM

▸ Stability:

▸ Constraints from the experiments testing additional long range force by light mediator with sensitivity

▸ Relic Density of relaxion:

▸ For relaxion to be DM:

Constraints

Relaxion DM

10-11 10-9 10-7 10-5 10-3 10-110111013101510171019 Fifth force

�V = (HI ) 4

max

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⇤br >

v

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

Tra=

150GeV

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AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k/9z2TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyrGnu0=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k/9z2TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyrGnu0=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k/9z2TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyrGnu0=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k/9z2TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyrGnu0=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k/9z2TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyrGnu0=

15GeV

AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k3zn3TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyxwnu4=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k3zn3TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyxwnu4=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k3zn3TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyxwnu4=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k3zn3TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyxwnu4=AAACE3icbVDNSgMxGMz6W+tf1aOXYCl4kLIrinoretBjBbctdJeSzWbb0CS7JtliWfoaetX38CRefQBfwycwbfdgWwcCw8z3fRkmSBhV2ra/raXlldW19cJGcXNre2e3tLffUHEqMXFxzGLZCpAijAriaqoZaSWSIB4w0gz6N2O/OSBS0Vg86GFCfI66gkYUI20k3zn3TjJPcnhLGqNOqWxX7QngInFyUgY56p3SjxfGOOVEaMyQUm3HTrSfIakpZmRUrHipIgnCfdQlbUMF4kT52ST1CFaMEsIoluYJDSdq8c9GhrhSQx6YSY50T817Y/E/r53q6NLPqEhSTQSefhSlDOoYjiuAIZUEazY0BGFJTViIe0girE1RM5fCAU1UnvppGtuU5MxXskjc0+pV1bk/K9eu87YK4BAcgWPggAtQA3egDlyAwSN4Aa/gzXq23q0P63M6umTlOwdgBtbXLyxwnu4=

Relaxion DM

10-11 10-9 10-7 10-5 10-3 10-110-2510-2310-2110-1910-1710-1510-1310-11

Unstable� ! XX

AAAB8XicdVBNS8NAFHzxs9avqkcvi0XwVJK21HorevFYwbbBJpTNdtMu3WzC7kYoof/CiwdFvPpvvPlv3LQVVHRgYZh5j30zQcKZ0rb9Ya2srq1vbBa2its7u3v7pYPDropTSWiHxDyWboAV5UzQjmaaUzeRFEcBp71gcpX7vXsqFYvFrZ4m1I/wSLCQEayNdOclY+bpGLnuoFS2K7bdrDUv0JzUG9UFaTRqyDEkRxmWaA9K794wJmlEhSYcK9V37ET7GZaaEU5nRS9VNMFkgke0b6jAEVV+Nr94hk6NMkRhLM0TGs3V7xsZjpSaRoGZjLAeq99eLv7l9VMdNv2MiSTVVJDFR2HKkcmYx0dDJinRfGoIJpKZWxEZY4mJNiUVTQlfSdH/pFutOLVK9aZebl0u6yjAMZzAGThwDi24hjZ0gICAB3iCZ0tZj9aL9boYXbGWO0fwA9bbJ4dKkNU=

Fifth force ⇤br

� vAAACAHicdVDLSsNAFJ3UV62vqAsXbgaL4CoktfSxK7px4aKCfUATwmQyaYdOHs5MCiV046+4caGIWz/DnX/jpK2gogcGDuecy517vIRRIU3zQyusrK6tbxQ3S1vbO7t7+v5BV8Qpx6SDYxbzvocEYTQiHUklI/2EExR6jPS88WXu9yaECxpHt3KaECdEw4gGFCOpJFc/sq9V2EduZvMQenwG7SG5gxNXL5tGs2ZVTAvmpFa1qjlpmPVmBVqGOUcZLNF29Xfbj3EakkhihoQYWGYinQxxSTEjs5KdCpIgPEZDMlA0QiERTjY/YAZPleLDIObqRRLO1e8TGQqFmIaeSoZIjsRvLxf/8gapDBpORqMklSTCi0VByqCMYd4G9CknWLKpIghzqv4K8QhxhKXqrKRK+LoU/k+6FcM6Nyo31XLrYllHERyDE3AGLFAHLXAF2qADMJiBB/AEnrV77VF70V4X0YK2nDkEP6C9fQJ9LJZb

f =Mp

l

AAAB+HicdZDLSgMxFIYz9VbrpaMu3QSL4KpML9S6EIpu3AgV7AXaYcikmTY0yQxJRqjDPIkbF4q49VHc+TZm2goq+kPg4z/ncE5+P2JUacf5sHIrq2vrG/nNwtb2zm7R3tvvqjCWmHRwyELZ95EijArS0VQz0o8kQdxnpOdPL7N6745IRUNxq2cRcTkaCxpQjLSxPLsYwHM4lDy59pKIpalnl5yy4zRrzTM4h3qjuoBGowYrBjKVwFJtz34fjkIccyI0ZkipQcWJtJsgqSlmJC0MY0UihKdoTAYGBeJEucn88BQeG2cEg1CaJzScu98nEsSVmnHfdHKkJ+p3LTP/qg1iHTTdhIoo1kTgxaIgZlCHMEsBjqgkWLOZAYQlNbdCPEESYW2yKpgQvn4K/4dutVyplas39VLrYhlHHhyCI3ACKuAUtMAVaIMOwCAGD+AJPFv31qP1Yr0uWnPWcuYA/JD19gmvQ5Mi

Relaxion Phenomenology

• It can have both scalar and pseudo-scalar coupling to the SM

• DM field with small mass ϕ mϕ ≲ 10−4 eV ⇔ ωϕ ≲ 103 MHz

Scalar:

ℒ ⊃ geϕ ēe +gγ4

ϕFμνFμν

1) Fifth force experiments e.g. EP test,…2) Oscillation of fundamental constants

1) Nuclear Magnetic Resonance2) Resonant magnetic cavity

me → me + ge ϕ(t)α → α + gγ ϕ(t)

Probed by atomic clock experiments, nuclear transitions…..

Pseudo-scalar:ℒ ⊃

gϕγγ4

ϕF̃μνFμν +igd2

ϕN̄ σμνγ5 NFμν + gϕNN∂μϕN̄ γμγ5 N

Probed by GNOME, CASPEr, ABRACADABRA……

“Axion-like”“Dilaton-like”

“Relaxion”

➤ Achieving incredible sensitivity to variation of constants!

➤ For background DM density, this corresponds to

➤ Constraints only at low mass/ frequency.

ge ≲ 10−20 ( mϕ10−15 eV ) gγ ≲ 10−18 GeV−1 ( mϕ10−15 eV )

( δmeme )exp

≃ 10−14 ( δαα )exp ≃ 10−16(Today)

( δmeme )exp

≃ 10−18 ( δαα )exp ≃ 10−18(Near future)

(mϕ ≲ 10−15 eV ∼ 1 Hz)

Atomic Clocks: New Frontier

See e.g. Aharony et al. (1902.02788)

No Longer

ℒ ⊃ ge ϕ ē e +gγ4

ϕ FμνFμν

ϕ(t) =2ρlocalmϕ

cos(mϕt)

δmeme

=ge ϕme

δαα

= gγϕ

See e.g. Arvanitaki et al. PRD91, 015015

➤ 5th force experiments put strong constraints.

➤ Any hope left ?

Arvanitaki et al. PRD91, 015015 Hees et al. 1807.04512

10-21 10-18 10-15 10-12 10-910-3110-2610-2110-1610-11

10-21 10-18 10-15 10-12 10-910-3110-2610-2110-1610-11

5th Force Constraints: Hard to Beat

➤ Light scalar field forms substructure.

➤ Boson star around us!

➤ Substructure described by ground state of classical, non-relativistic field

➤ Light scalar DM suppresses gravitationally-bound substructure at scales smaller than

➤ Boson star doesn’t need self-gravity to be stable

Local Density vs Background Density

Eby, Leembruggen, Suranyi, Wijewardhana (1608.06911)

Kolb and Tkachev (PRL 1993), Schive, Chiueh, Broadhurst (1406.6586)

Levkov, Panin, Tkachev (1804.05857) Vaquero, Redondo, Stadler (1809.09241)

………

λDB ≈ R⊕ ( 10−11eVmϕ ) (

10−3

v )

Possible we live in a (very) local over density of Relaxion Halo

Large scalar field density allowed, if bound to the Earth or the Sun

SCALAR DM SUBSTRUCTURE

1) Boson stars - Self Gravitating

2) Relaxion Halos - External Gravity

galaxysolar system

boson starboson star

sun

R? /1

M?AAACDnicbVDLSgMxFM34rPU16tJNsBRclZkq6LLoxo1QxT6gMwyZNNOGZjIhyQhlmC9w46+4caGIW9fu/BvTdgRtPRA4nHMvN+eEglGlHefLWlpeWV1bL22UN7e2d3btvf22SlKJSQsnLJHdECnCKCctTTUjXSEJikNGOuHocuJ37olUNOF3eiyIH6MBpxHFSBspsKu3QeYpjWQOPSEToRPoRRLhzM2z6x8rD+yKU3OmgIvELUgFFGgG9qfXT3AaE64xQ0r1XEdoP0NSU8xIXvZSRQTCIzQgPUM5ionys2mcHFaN0odRIs3jGk7V3xsZipUax6GZjJEeqnlvIv7n9VIdnfsZ5SLVhOPZoShl0ISedAP7VBKs2dgQhCU1f4V4iEwb2jRYNiW485EXSbtec09q9ZvTSuOiqKMEDsEROAYuOAMNcAWaoAUweABP4AW8Wo/Ws/Vmvc9Gl6xi5wD8gfXxDV6JnPM=

R? independent of M?AAACFnicdVDLSgMxFM34rPVVdekmWAQ3lmm1tuKm6MaNUMWq0BlKJr2jwcyD5I5Yhn6FG3/FjQtF3Io7/8a0U0FFLyQczrk3ued4sRQabfvDGhufmJyazs3kZ+fmFxYLS8tnOkoUhxaPZKQuPKZBihBaKFDCRayABZ6Ec+/6YKCf34DSIgpPsReDG7DLUPiCMzRUp7B50kkdjUz1nT3qINxiKsIuxGCuEGnkD/ijr55OoWiXKtXdHbtOM1CtZcBAWi7ZwyqSUTU7hXenG/EkMG9xybRul+0Y3ZQpFFxCP+8kGmLGr9kltA0MWQDaTYe2+nTdMF3qR8ocs8uQ/T6RskDrXuCZzoDhlf6tDci/tHaCft01PuMEIeTZR34iKUZ0kBHtCgUcZc8AxpUwu1J+xRTjaJLMmxC+nNL/wVmlVN4qVY63i439URw5skrWyAYpkxppkEPSJC3CyR15IE/k2bq3Hq0X6zVrHbNGMyvkR1lvn6qloFg=

Relaxion HaloEarth

DM Background

δ ≫ 1

Halo supported by Sun “Solar Halo”

Sun

mϕ = 10−17 ÷ 10−13 eV∼ mHz ÷ 10 Hz

Relaxion Halo

Earth

DM Background

δ ≫ 1

Halo supported by Earth “Earth Halo”

mϕ = 10−13 ÷ 10−8 eV∼ 10 Hz ÷ MHz

R⋆ > RER⋆ > AU

R⋆ ≈M2P

m2ϕ Mext

Relaxion Halo Slide by Josh Eby

Enhancement 1: Amplitude

Signal ∝ ϕ

Large field values possible!

AB, H Kim, G. Perez, D. Budkar, J. Eby, O. Matsedonskii (1912.04295)

(contours : M⋆/Mext)

Enhancement 2: COHERENCE TIME

τϕ ≃ 103 s ( 10−9 eVmϕ )

3

(Earth Halo)

τϕ ≃ 1010 s ( 10−15 eVmϕ )

3

(Solar Halo)

http://arxiv.org/abs/arXiv:1912.04295

Enhancement 3: Field GRADIENTAB, H Kim, G. Perez, D. Budkar, J. Eby, O. Matsedonskii (1912.04295)

Some signals ∝ ⃗∇ ϕ

Halo is like a gravitational atom

ϕ(r) ∝ exp (−r/R⋆)⃗∇ ϕ ∝ ̂r

R⋆“radial gradient”

⃗∇ ϕ ∝ mϕ ⃗v rel“tangential gradient”

It also has a “wind” part:

Dominant in low mass range!

(contours : M⋆ /Mext)

http://arxiv.org/abs/arXiv:1912.04295

Possible enhancement factors - Amplitude, Coherence time, velocity

All Hail to Halo!

All Hail to Halo - 2 !

Scalar:ℒ ⊃ geϕ ēe +

gγ4

ϕFμνFμν

1) Fifth force experiments e.g. EP test,…2) Oscillation of fundamental constants 1) Neutron EDM

2) Spin dependent couplings3) ALP Photon interaction

Probed by atomic clock experiments, nuclear transition…..

Pseudo-scalar:

Probed by GNOME, CASPEr, ABRACADABRA……

“Axion”“Dilaton”

“Relaxion”

δmeme

=ge ϕme

δαα

= gγϕ

ℒ ⊃gϕγγ

4ϕF̃μνFμν +

igd2


Coming Back! How does Halo improve these experiments??

10-20 10-17 10-14 10-11 10-810-2910-2710

-2510-2310-2110

-19

10-20 10-17 10-14 10-11 10-810-2610-2410-2210-2010-1810-16

10-20 10-17 10-14 10-11 10-810-2910-2710

-2510-2310-2110

-19

10-20 10-17 10-14 10-11 10-810-2610-2410-2210-2010-1810-16

Solar HaloSolid: Dashed: Dotted:10−14 10−16 10−18Sensitivity:

Earth Halo

Enhancement in Halo scenario - 1 δmeme = ge ϕmeδαα

= gγϕ

CONCLUSIONS

CONCLUSIONS

➤ Relaxion which addresses Hierarchy Problem can be identified as DM.

CONCLUSIONS


➤ Relaxion could be identified as light DM which has both scalar and pseudo-scalar couplings.

CONCLUSIONS



➤ Atomic physics experiments to probe scalar DM.

CONCLUSIONS




➤ Relaxion halo in the solar system gives a unique handle for detection.

CONCLUSIONS





➤ Huge sensitivity enhancement in halo scenario.

CONCLUSIONS






➤ Halo formation story (!)

CONCLUSIONS







➤ Pseudo-scalar coupling: Halo scenario also modifies sensitivity for those.

CONCLUSIONS







➤ Pseudo-scalar coupling: Halo scenario also modifies sensitivity for those.

➤ CASPEr, ABRACADABRA, GNOME, DM Radio wait for Josh!

תודה

RELAXION - 100

UNCUT IT !

‣ Relaxion Potential:

‣ Higgs Mass Parameter:

‣ During Inflation changes from positive to negative.

‣ When, back-reaction appears and stops the relaxion motion:

‣ Minimum Mass:

REVIEW: RELAXION MECHANISM

!(#)

#

&' # >0

&' #

COSMOLOGICAL HISTORY ▸ Reheating Temperature higher than

barrier height Wiggles will disappear.

▸ Relaxion starts to roll down again

▸ Universe cools down wiggles reappear .

▸ If relaxion can be trapped again it will have some displacement from the minimum.

▸ Hubble drops below the mass of relaxion It will start to oscillate.

!(#)

#

30

T = 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◆4 ✓100GeVTosc

◆3

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

Displacement from the Minima

Height of the barrier Potential

Time when it starts to Oscillate

⌦DMh2 = 0.12

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

‣ Coherently oscillating Relaxion field could constitutes DM !!

‣ Observed Density:

Relaxion DM

CONCLUSION

The relaxion is light because it is axion-like particle but it mixes with the Higgs - has scalar interactions.

Relaxion could be identified as coherent dark matter.

Can apply axion dark matter searches

Fundamental parameters in the SM, will oscillate.

Flacke, Frugiuele, Fuchs, Gupta & GP; Choi & Im (16)

SCALES MATTER FOR SCALAR MATTER

➤ Small scales: Light scalar DM suppresses gravitationally-bound substructure at scales smaller than

➤ Substructure described by ground state of classical field

➤ “Boson star”, “Axion condensate”, “Soliton”, “Oscillaton”, …

λdB = 1 AU × ( 10−15 eVm ) ( 10

−3cv )





λdB = 1 AU × ( 10−15 eVm ) ( 10

−3cv )

Two classes of questions:

What kinds of boson stars are around?

(Formation, History, Spectrum)

If boson stars are around…

What can we search for?

(Stability, Interactions, Signals)Most of this talk



………

MASS-RADIUS RELATION➤ Possible we live in a (very) local over/underdensity

➤ Intriguing that scalar field bound to external object has stable ground-state configuration with size independent of !

➤ only constrained by measurements of local gravity

R⋆ ≈M2P

m2ϕ Mext

34

M⋆

i ·ψ = [− ∇2

2m+ Vg+Vext] ψ Mext ≫ M⋆

M⋆

➤ Relaxion halo is coherently oscillating state ➤ Coherence length

➤ Coherence time

Rcoh =1

mϕ v=

M2Pm2ϕ Mext

= R⋆

τ⋆ =1

mϕ v2≈ {

103 sec × (10−9 eV/mϕ)3 for R⋆ > R⊕103 sec for R⋆ ≲ R⊕

(for Earth halo)

10-18 10-16 10-14 10-12 10-10 10-8 10-610-1410-1210-1010-810-610-4

Earth HaloSun HaloALP DM 10-18 10-16 10-14 10-12 10-10 10-8 10-610-8

10-610-410-2100102

Earth HaloSun HaloALP DM

Halo Coherence Length & Time


10-17 10-15 10-13 10-11 10-9 10-710-1210-1010-810-610-410-2100

Adler (0808.0899)Pitjev and Pitjeva (1306.5534)

Lunar Laser Ranging

+ LAGEOS Satellite

Solar System Ephemerides

(Mercury, Mars, Saturn)

36

➤ Constraint on ge, gγ ⇔ ρ⋆ ∝ M⋆

M⋆

Constraints on Local Over Density

THE “LOCAL DENSITY”➤ Need to think outside the box.

Standard Picture “Galaxy Halo”

37

Earth

DM Background

As everybody knows,ρlocal = 0.4 GeV/cm

3

➤ We know that collisions with self-gravitating boson stars are rare


➤ Large scalar field density allowed, if bound to Earth or Sun

GC

Assumes viral equilibrium of galaxy

MISALIGNMENT MECHANISM

‣ Consider a scalar field in expanding universe with potential

‣ EOM:

‣ When, , then solves EOM.

‣ When, ,

‣ Oscillation frequency greater than the expansion rate.

-4 -2 2 4ϕ

2

4

6

8

10

12

V(ϕ)

Initial Misalignment

V(ϕ)

ϕ

➤ Can do better than fifth-force tests, probing natural models near

10-20 10-17 10-14 10-11 10-810-2910-2710

-2510-2310-2110

-19

10-20 10-17 10-14 10-11 10-810-2610-2410-2210-2010-1810-16

Solid: Dashed: 10−16 10−18‣ Sensitivity:

mϕ ∼ 10−16 eV

Constraints on Solar Halo

➤ Can even probe physical relaxion models in near future!

10-20 10-17 10-14 10-11 10-810-2910-2710

-2510-2310-2110

-19

10-20 10-17 10-14 10-11 10-810-2610-2410-2210-2010-1810-16

Solid: Dashed: Dotted:10−14 10−16 10−18‣ Sensitivity:

Constraints on Earth Halo

➤ Relaxion breaks CP, mixes with Higgs

➤ Scalar + pseudo-scalar couplings

41

10-22 10-17 10-12 10-7 0.01105

109

1013

1017

m [eV]

f[GeV

]

� > ��

�-�

>��

�� (��)

��

(��)

��

��

��

��

��

�/ ��

��

(��)

Plot by Josh Eby


Oscillation of fundamental constants

How to Find them??



10-22 10-17 10-12 10-7 0.01105

109

1013

1017

m [eV]

f[GeV

]

� > ��

�-�

>��

�� (��)

��

(��)

��

��

��

��

��

�/ ��

��

(��)




ϕ FμνFμν

δme⟨me⟩

=geϕ⟨me⟩

δαα

= gγϕ

ϕ(t) =2ρlocalmϕ

cos(mϕt)

How to Find it??

Plot by Josh Eby



10-22 10-17 10-12 10-7 0.01105

109

1013

1017

m [eV]

f[GeV

]

� > ��

�-�

>��

�� (��)

��

(��)

��

��

��

��

��

�/ ��

��

(��)




ϕ FμνFμν

δme⟨me⟩

=geϕ⟨me⟩

δαα

= gγϕ

ϕ(t) =2ρlocalmϕ

cos(mϕt)

How to Detect it??

Atomic Clock Experiments

Plot by Josh Eby



10-22 10-17 10-12 10-7 0.01105

109

1013

1017

m [eV]

f[GeV

]

� > ��

�-�

>��

�� (��)

��

(��)

��

��

��

��

��

�/ ��

��

(��)




ϕ FμνFμν

δme⟨me⟩

=geϕ⟨me⟩

δαα

= gγϕ

ϕ(t) =2ρlocalmϕ

cos(mϕt)

How to Detect it??

Atomic Clock Experiments See e.g. Arvanitaki et al. PRD91, 015015

Plot by Josh Eby

➤ Relaxion halo is coherently oscillating state ➤ Coherence length

➤ Coherence time

Rcoh =1

mϕ v=

M2Pm2ϕ Mext

= R⋆

τ⋆ =1

mϕ v2≈ {

103 sec × (10−9 eV/mϕ)3 for R⋆ > R⊕103 sec for R⋆ ≲ R⊕

(for Earth halo)

10-18 10-16 10-14 10-12 10-10 10-8 10-610-1410-1210-1010-810-610-4

Earth HaloSun HaloALP DM 10-18 10-16 10-14 10-12 10-10 10-8 10-610-8

10-610-410-2100102

Earth HaloSun HaloALP DM

Halo Coherence Length & Time





λDB ≈ R⊕ ( 10−11eVmϕ ) (

10−3

v )








If boson stars are around…

What can we search for?

(Stability, Interactions, Signals)Most of this talk



………

λDB ≈ R⊕ ( 10−11eVmϕ ) (

10−3

v )

Capturedby Earth

by Sun10-18 10-16 10-14 10-12 10-10 10-8 10-610-4210-3710-3210-2710-2210-17

Halo Over Density



− ρlocal = 0.4 GeV/cm3

Earth

DM Background

GC


At leading order, - radially symmetric

− vvir ∼ 200 km/s


Eby, Leembruggen, Suranyi, Wijewardhana (1608.06911)

➤ We know that collisions with self-gravitating boson stars are rare


➤ Large scalar field density allowed, if bound to Earth or Sun

➤ Possible we live in a (very) local over/under-density


− ρlocal = 0.4 GeV/cm3

Earth

DM Background

GC


At leading order, - radially symmetric

− vvir ∼ 200 km/s

MASS-RADIUS RELATION

➤ Intriguing that scalar field bound to external object has stable ground-state configuration with size independent of !


M⋆

M⋆

Mext ≫ M⋆ R⋆ ≈M2P

m2ϕ Mexti

∂∂t

ψ = [− ∇2

2m+Vg( |ψ |

2 ) + Vext( |ψ |2 ) + Vint( |ψ |

2 )] ψTime dependence

Kinetic Energy Self interaction

Self and external gravity

NR classical field, oscillates with ω ∼ mϕBalance these forces


10-17 10-15 10-13 10-11 10-9 10-710-1210-1010-810-610-410-2100

Adler (0808.0899)Pitjev and Pitjeva (1306.5534)

Lunar Laser Ranging

+ LAGEOS Satellite

Solar System Ephemerides

(Mercury, Mars, Saturn)

➤ Constraint on ge, gγ ⇔ ρ⋆ ∝ M⋆

M⋆

Constraints on Local Over-Density

Pseudo-scalar Couplings

ℒ ⊃gϕγγ

4ϕF̃μνFμν +

igd2


Modifies Maxwell’s eq. and induces an “effective current”

⃗jeff = gϕγγ ⃗B ∂ϕ∂t

Probed by, ABRACADABRA

Hwind = − gϕNN ⃗∇ ϕ ⋅ ⃗σ

In the NR limit, it gives “effective” magnetic field

Probed by, CASPEr Wind and GNOME

Probed by, CASPEr Electric

In the NR limit, it gives Neutron EDM

HEDM = − gdϕ ⃗σ ⋅ ⃗E⃗d N

} }

γN ⃗B ϕ(t)

Enhancement in Halo scenario - 2Enhancement from Amplitude and Coherence time

HEDM = − gϕNNϕ ⃗σ ⋅ ⃗E = − ⃗d N(t) ⋅ ⃗E ⃗jeff(t) = gϕγγmϕ ϕ cos(mϕt) ⃗B

ABRACADABRA

10-16 10-14 10-12 10-10 10-8 10-6 10-410-30

10-2510-2010-1510-10

CASPEr Electric

10-16 10-14 10-12 10-10 10-8 10-610-3510-3010

-2510-2010

-1510-10

Enhancement in Halo scenario - 3

Enhancement from amplitude and gradient

HEDM = − gd ⃗∇ ϕ ⋅ ⃗σ = − ⃗μ N ⋅ ⃗Bϕ (t)

CASPEr Wind

10-17 10-15 10-13 10-11 10-9 10-710-710-510-310-1101

GNOME

10-17 10-16 10-15 10-14 10-1310-1310-1210-1110-1010-910-8

DARK MATTER ?

NATURALNESS PROBLEM - 102

▸ The Standard Model of ParticlePhysics is valid up to some scale

▸ Higgs mass:

▸ In the SM:

▸ There is a cancellation of order

▸ It requires an explanation!!

Not So Correct!

Enough!PACK IT !

EXTRA 56

FRICTION FROM HUBBLE: MINIMAL SCENARIO

▸ Field Velocity :

▸ Relaxion will be trapped:

▸ Misalignment angle

▸ This case:

▸ Overabundance of DM - New constraint on any back-reaction Model !!

57

�̇(H) =1

5

⇤4brfH

AAACIHicbZDLSgMxFIYz9VbrbdSlm2AR6qbMSKVuhKKbLlxUsBfojEMmzbShmQtJRihhHsWNr+LGhSK606cxbWehrQcCH/+55fx+wqiQlvVlFFZW19Y3ipulre2d3T1z/6Aj4pRj0sYxi3nPR4IwGpG2pJKRXsIJCn1Guv74eprvPhAuaBzdyUlC3BANIxpQjKSWPLMOnUEslZOMaFZpnsJL6AQcYWVn6jybo3Ojxw2Qp3ye3dcyFcBmBj2zbFWtWcBlsHMogzxanvmpF+E0JJHEDAnRt61EugpxSTEjWclJBUkQHqMh6WuMUEiEq2YHZvBEKwMYxFy/SMKZ+rtDoVCISejryhDJkVjMTcX/cv1UBheuolGSShLh+aIgZVDGcOoWHFBOsGQTDQhzqv8K8QhpV6T2tKRNsBdPXobOWdXWfFsrN65yO4rgCByDCrBBHTRAE7RAG2DwCJ7BK3gznowX4934mJcWjLznEPwJ4/sHxsyivQ==AAACIHicbZDLSgMxFIYz9VbrbdSlm2AR6qbMSKVuhKKbLlxUsBfojEMmzbShmQtJRihhHsWNr+LGhSK606cxbWehrQcCH/+55fx+wqiQlvVlFFZW19Y3ipulre2d3T1z/6Aj4pRj0sYxi3nPR4IwGpG2pJKRXsIJCn1Guv74eprvPhAuaBzdyUlC3BANIxpQjKSWPLMOnUEslZOMaFZpnsJL6AQcYWVn6jybo3Ojxw2Qp3ye3dcyFcBmBj2zbFWtWcBlsHMogzxanvmpF+E0JJHEDAnRt61EugpxSTEjWclJBUkQHqMh6WuMUEiEq2YHZvBEKwMYxFy/SMKZ+rtDoVCISejryhDJkVjMTcX/cv1UBheuolGSShLh+aIgZVDGcOoWHFBOsGQTDQhzqv8K8QhpV6T2tKRNsBdPXobOWdXWfFsrN65yO4rgCByDCrBBHTRAE7RAG2DwCJ7BK3gznowX4934mJcWjLznEPwJ4/sHxsyivQ==AAACIHicbZDLSgMxFIYz9VbrbdSlm2AR6qbMSKVuhKKbLlxUsBfojEMmzbShmQtJRihhHsWNr+LGhSK606cxbWehrQcCH/+55fx+wqiQlvVlFFZW19Y3ipulre2d3T1z/6Aj4pRj0sYxi3nPR4IwGpG2pJKRXsIJCn1Guv74eprvPhAuaBzdyUlC3BANIxpQjKSWPLMOnUEslZOMaFZpnsJL6AQcYWVn6jybo3Ojxw2Qp3ye3dcyFcBmBj2zbFWtWcBlsHMogzxanvmpF+E0JJHEDAnRt61EugpxSTEjWclJBUkQHqMh6WuMUEiEq2YHZvBEKwMYxFy/SMKZ+rtDoVCISejryhDJkVjMTcX/cv1UBheuolGSShLh+aIgZVDGcOoWHFBOsGQTDQhzqv8K8QhpV6T2tKRNsBdPXobOWdXWfFsrN65yO4rgCByDCrBBHTRAE7RAG2DwCJ7BK3gznowX4934mJcWjLznEPwJ4/sHxsyivQ==AAACIHicbZDLSgMxFIYz9VbrbdSlm2AR6qbMSKVuhKKbLlxUsBfojEMmzbShmQtJRihhHsWNr+LGhSK606cxbWehrQcCH/+55fx+wqiQlvVlFFZW19Y3ipulre2d3T1z/6Aj4pRj0sYxi3nPR4IwGpG2pJKRXsIJCn1Guv74eprvPhAuaBzdyUlC3BANIxpQjKSWPLMOnUEslZOMaFZpnsJL6AQcYWVn6jybo3Ojxw2Qp3ye3dcyFcBmBj2zbFWtWcBlsHMogzxanvmpF+E0JJHEDAnRt61EugpxSTEjWclJBUkQHqMh6WuMUEiEq2YHZvBEKwMYxFy/SMKZ+rtDoVCISejryhDJkVjMTcX/cv1UBheuolGSShLh+aIgZVDGcOoWHFBOsGQTDQhzqv8K8QhpV6T2tKRNsBdPXobOWdXWfFsrN65yO4rgCByDCrBBHTRAE7RAG2DwCJ7BK3gznowX4934mJcWjLznEPwJ4/sHxsyivQ==

�̇2(H) . ⇤4br

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✓ ⌘ �/fAAAB/XicbZDJSgNBEIZ7XGPcxuXmpTEInuKMCHoMevEYwSyQGUJPpyZp0rPYXROIQ/BVvHhQxKvv4c23sbMcNPGHho+/qqjqP0il0Og439bS8srq2npho7i5tb2za+/t13WSKQ41nshENQOmQYoYaihQQjNVwKJAQiPo34zrjQEoLZL4Hocp+BHrxiIUnKGx2vahhz1ARj14yMSAemlPnIVtu+SUnYnoIrgzKJGZqm37y+skPIsgRi6Z1i3XSdHPmULBJYyKXqYhZbzPutAyGLMItJ9Prh/RE+N0aJgo82KkE/f3RM4irYdRYDojhj09Xxub/9VaGYZXfi7iNEOI+XRRmEmKCR1HQTtCAUc5NMC4EuZWyntMMY4msKIJwZ3/8iLUz8uu4buLUuV6FkeBHJFjckpcckkq5JZUSY1w8kieySt5s56sF+vd+pi2LlmzmQPyR9bnD+yKlN0=AAAB/XicbZDJSgNBEIZ7XGPcxuXmpTEInuKMCHoMevEYwSyQGUJPpyZp0rPYXROIQ/BVvHhQxKvv4c23sbMcNPGHho+/qqjqP0il0Og439bS8srq2npho7i5tb2za+/t13WSKQ41nshENQOmQYoYaihQQjNVwKJAQiPo34zrjQEoLZL4Hocp+BHrxiIUnKGx2vahhz1ARj14yMSAemlPnIVtu+SUnYnoIrgzKJGZqm37y+skPIsgRi6Z1i3XSdHPmULBJYyKXqYhZbzPutAyGLMItJ9Prh/RE+N0aJgo82KkE/f3RM4irYdRYDojhj09Xxub/9VaGYZXfi7iNEOI+XRRmEmKCR1HQTtCAUc5NMC4EuZWyntMMY4msKIJwZ3/8iLUz8uu4buLUuV6FkeBHJFjckpcckkq5JZUSY1w8kieySt5s56sF+vd+pi2LlmzmQPyR9bnD+yKlN0=AAAB/XicbZDJSgNBEIZ7XGPcxuXmpTEInuKMCHoMevEYwSyQGUJPpyZp0rPYXROIQ/BVvHhQxKvv4c23sbMcNPGHho+/qqjqP0il0Og439bS8srq2npho7i5tb2za+/t13WSKQ41nshENQOmQYoYaihQQjNVwKJAQiPo34zrjQEoLZL4Hocp+BHrxiIUnKGx2vahhz1ARj14yMSAemlPnIVtu+SUnYnoIrgzKJGZqm37y+skPIsgRi6Z1i3XSdHPmULBJYyKXqYhZbzPutAyGLMItJ9Prh/RE+N0aJgo82KkE/f3RM4irYdRYDojhj09Xxub/9VaGYZXfi7iNEOI+XRRmEmKCR1HQTtCAUc5NMC4EuZWyntMMY4msKIJwZ3/8iLUz8uu4buLUuV6FkeBHJFjckpcckkq5JZUSY1w8kieySt5s56sF+vd+pi2LlmzmQPyR9bnD+yKlN0=AAAB/XicbZDJSgNBEIZ7XGPcxuXmpTEInuKMCHoMevEYwSyQGUJPpyZp0rPYXROIQ/BVvHhQxKvv4c23sbMcNPGHho+/qqjqP0il0Og439bS8srq2npho7i5tb2za+/t13WSKQ41nshENQOmQYoYaihQQjNVwKJAQiPo34zrjQEoLZL4Hocp+BHrxiIUnKGx2vahhz1ARj14yMSAemlPnIVtu+SUnYnoIrgzKJGZqm37y+skPIsgRi6Z1i3XSdHPmULBJYyKXqYhZbzPutAyGLMItJ9Prh/RE+N0aJgo82KkE/f3RM4irYdRYDojhj09Xxub/9VaGYZXfi7iNEOI+XRRmEmKCR1HQTtCAUc5NMC4EuZWyntMMY4msKIJwZ3/8iLUz8uu4buLUuV6FkeBHJFjckpcckkq5JZUSY1w8kieySt5s56sF+vd+pi2LlmzmQPyR9bnD+yKlN0=

m� =⇤2brf

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Very small !!

⌦�h2 / m9/2f2

AAACDHicbVC7TsMwFHV4lvIqMLJYVEhMJYmQgK2ChY0i0YfUpJXjOo1VO7FsB6mK8gEs/AoLAwix8gFs/A1umwFajmTp6JxzdX1PIBhV2ra/raXlldW19dJGeXNre2e3srffUkkqMWnihCWyEyBFGI1JU1PNSEdIgnjASDsYXU/89gORiibxvR4L4nM0jGlIMdJG6leq3i0nQ9TPPBHRPOq50BMyETqBvJddnrp52HNNyq7ZU8BF4hSkCgo0+pUvb5DglJNYY4aU6jq20H6GpKaYkbzspYoIhEdoSLqGxogT5WfTY3J4bJQBDBNpXqzhVP09kSGu1JgHJsmRjtS8NxH/87qpDi/8jMYi1STGs0VhyqC5ddIMHFBJsGZjQxCW1PwV4ghJhLXpr2xKcOZPXiQtt+YYfndWrV8VdZTAITgCJ8AB56AObkADNAEGj+AZvII368l6sd6tj1l0ySpmDsAfWJ8/0TOazQ==AAACDHicbVC7TsMwFHV4lvIqMLJYVEhMJYmQgK2ChY0i0YfUpJXjOo1VO7FsB6mK8gEs/AoLAwix8gFs/A1umwFajmTp6JxzdX1PIBhV2ra/raXlldW19dJGeXNre2e3srffUkkqMWnihCWyEyBFGI1JU1PNSEdIgnjASDsYXU/89gORiibxvR4L4nM0jGlIMdJG6leq3i0nQ9TPPBHRPOq50BMyETqBvJddnrp52HNNyq7ZU8BF4hSkCgo0+pUvb5DglJNYY4aU6jq20H6GpKaYkbzspYoIhEdoSLqGxogT5WfTY3J4bJQBDBNpXqzhVP09kSGu1JgHJsmRjtS8NxH/87qpDi/8jMYi1STGs0VhyqC5ddIMHFBJsGZjQxCW1PwV4ghJhLXpr2xKcOZPXiQtt+YYfndWrV8VdZTAITgCJ8AB56AObkADNAEGj+AZvII368l6sd6tj1l0ySpmDsAfWJ8/0TOazQ==AAACDHicbVC7TsMwFHV4lvIqMLJYVEhMJYmQgK2ChY0i0YfUpJXjOo1VO7FsB6mK8gEs/AoLAwix8gFs/A1umwFajmTp6JxzdX1PIBhV2ra/raXlldW19dJGeXNre2e3srffUkkqMWnihCWyEyBFGI1JU1PNSEdIgnjASDsYXU/89gORiibxvR4L4nM0jGlIMdJG6leq3i0nQ9TPPBHRPOq50BMyETqBvJddnrp52HNNyq7ZU8BF4hSkCgo0+pUvb5DglJNYY4aU6jq20H6GpKaYkbzspYoIhEdoSLqGxogT5WfTY3J4bJQBDBNpXqzhVP09kSGu1JgHJsmRjtS8NxH/87qpDi/8jMYi1STGs0VhyqC5ddIMHFBJsGZjQxCW1PwV4ghJhLXpr2xKcOZPXiQtt+YYfndWrV8VdZTAITgCJ8AB56AObkADNAEGj+AZvII368l6sd6tj1l0ySpmDsAfWJ8/0TOazQ==AAACDHicbVC7TsMwFHV4lvIqMLJYVEhMJYmQgK2ChY0i0YfUpJXjOo1VO7FsB6mK8gEs/AoLAwix8gFs/A1umwFajmTp6JxzdX1PIBhV2ra/raXlldW19dJGeXNre2e3srffUkkqMWnihCWyEyBFGI1JU1PNSEdIgnjASDsYXU/89gORiibxvR4L4nM0jGlIMdJG6leq3i0nQ9TPPBHRPOq50BMyETqBvJddnrp52HNNyq7ZU8BF4hSkCgo0+pUvb5DglJNYY4aU6jq20H6GpKaYkbzspYoIhEdoSLqGxogT5WfTY3J4bJQBDBNpXqzhVP09kSGu1JgHJsmRjtS8NxH/87qpDi/8jMYi1STGs0VhyqC5ddIMHFBJsGZjQxCW1PwV4ghJhLXpr2xKcOZPXiQtt+YYfndWrV8VdZTAITgCJ8AB56AObkADNAEGj+AZvII368l6sd6tj1l0ySpmDsAfWJ8/0TOazQ==

▸ Relic Density:

Ttr = Tra Tosc / m1/2� TraAAACG3icbVDLSgMxFM3UV62vUZdugkVwVWeKoMuiG5cV+oJOHTJp2oYmk5hkhDLMf7jxV9y4UMSV4MK/MW1noa0HAodzzuXmnkgyqo3nfTuFldW19Y3iZmlre2d3z90/aGmRKEyaWDChOhHShNGYNA01jHSkIohHjLSj8fXUbz8QpamIG2YiSY+jYUwHFCNjpdCtNsI0UBwKjTMYSCWkEZBbTY5odpf6Z1UrM3IP85xCWeiWvYo3A1wmfk7KIEc9dD+DvsAJJ7HBDGnd9T1peilShmJGslKQaCIRHqMh6VoaI050L53dlsETq/ThQCj7YgNn6u+JFHGtJzyySY7MSC96U/E/r5uYwWUvpbFMDInxfNEgYdDePy0K9qki2LCJJQgrav8K8QgphI2ts2RL8BdPXiatasW3/Pa8XLvK6yiCI3AMToEPLkAN3IA6aAIMHsEzeAVvzpPz4rw7H/NowclnDsEfOF8/rhOhKg==AAACG3icbVDLSgMxFM3UV62vUZdugkVwVWeKoMuiG5cV+oJOHTJp2oYmk5hkhDLMf7jxV9y4UMSV4MK/MW1noa0HAodzzuXmnkgyqo3nfTuFldW19Y3iZmlre2d3z90/aGmRKEyaWDChOhHShNGYNA01jHSkIohHjLSj8fXUbz8QpamIG2YiSY+jYUwHFCNjpdCtNsI0UBwKjTMYSCWkEZBbTY5odpf6Z1UrM3IP85xCWeiWvYo3A1wmfk7KIEc9dD+DvsAJJ7HBDGnd9T1peilShmJGslKQaCIRHqMh6VoaI050L53dlsETq/ThQCj7YgNn6u+JFHGtJzyySY7MSC96U/E/r5uYwWUvpbFMDInxfNEgYdDePy0K9qki2LCJJQgrav8K8QgphI2ts2RL8BdPXiatasW3/Pa8XLvK6yiCI3AMToEPLkAN3IA6aAIMHsEzeAVvzpPz4rw7H/NowclnDsEfOF8/rhOhKg==AAACG3icbVDLSgMxFM3UV62vUZdugkVwVWeKoMuiG5cV+oJOHTJp2oYmk5hkhDLMf7jxV9y4UMSV4MK/MW1noa0HAodzzuXmnkgyqo3nfTuFldW19Y3iZmlre2d3z90/aGmRKEyaWDChOhHShNGYNA01jHSkIohHjLSj8fXUbz8QpamIG2YiSY+jYUwHFCNjpdCtNsI0UBwKjTMYSCWkEZBbTY5odpf6Z1UrM3IP85xCWeiWvYo3A1wmfk7KIEc9dD+DvsAJJ7HBDGnd9T1peilShmJGslKQaCIRHqMh6VoaI050L53dlsETq/ThQCj7YgNn6u+JFHGtJzyySY7MSC96U/E/r5uYwWUvpbFMDInxfNEgYdDePy0K9qki2LCJJQgrav8K8QgphI2ts2RL8BdPXiatasW3/Pa8XLvK6yiCI3AMToEPLkAN3IA6aAIMHsEzeAVvzpPz4rw7H/NowclnDsEfOF8/rhOhKg==AAACG3icbVDLSgMxFM3UV62vUZdugkVwVWeKoMuiG5cV+oJOHTJp2oYmk5hkhDLMf7jxV9y4UMSV4MK/MW1noa0HAodzzuXmnkgyqo3nfTuFldW19Y3iZmlre2d3z90/aGmRKEyaWDChOhHShNGYNA01jHSkIohHjLSj8fXUbz8QpamIG2YiSY+jYUwHFCNjpdCtNsI0UBwKjTMYSCWkEZBbTY5odpf6Z1UrM3IP85xCWeiWvYo3A1wmfk7KIEc9dD+DvsAJJ7HBDGnd9T1peilShmJGslKQaCIRHqMh6VoaI050L53dlsETq/ThQCj7YgNn6u+JFHGtJzyySY7MSC96U/E/r5uYwWUvpbFMDInxfNEgYdDePy0K9qki2LCJJQgrav8K8QgphI2ts2RL8BdPXiatasW3/Pa8XLvK6yiCI3AMToEPLkAN3IA6aAIMHsEzeAVvzpPz4rw7H/NowclnDsEfOF8/rhOhKg==

&‣ Timeline:

FRICTION FROM DARK PHOTON

▸ Adding a term:

▸ Exponential production of gauge field

▸ Field velocity after PP:

▸ Misalignment angle

▸ Timeline:

58

and Tosc = TtrAAACAXicbZDLSgMxFIYz9VbrbdSN4CZYBFdlRgTdCEU3Liv0Bu0wZNJMG5rLkGSEMowbX8WNC0Xc+hbufBvTdhba+kPgy3/OITl/lDCqjed9O6WV1bX1jfJmZWt7Z3fP3T9oa5kqTFpYMqm6EdKEUUFahhpGuokiiEeMdKLx7bTeeSBKUymaZpKQgKOhoDHFyFgrdI+aYdZXHEqNc3gNi5tReehWvZo3E1wGv4AqKNQI3a/+QOKUE2EwQ1r3fC8xQYaUoZiRvNJPNUkQHqMh6VkUiBMdZLMNcnhqnQGMpbJHGDhzf09kiGs94ZHt5MiM9GJtav5X66UmvgoyKpLUEIHnD8Upg0bCaRxwQBXBhk0sIKyo/SvEI6QQNja0ig3BX1x5GdrnNd/y/UW1flPEUQbH4AScAR9cgjq4Aw3QAhg8gmfwCt6cJ+fFeXc+5q0lp5g5BH/kfP4AkJGWTw==AAACAXicbZDLSgMxFIYz9VbrbdSN4CZYBFdlRgTdCEU3Liv0Bu0wZNJMG5rLkGSEMowbX8WNC0Xc+hbufBvTdhba+kPgy3/OITl/lDCqjed9O6WV1bX1jfJmZWt7Z3fP3T9oa5kqTFpYMqm6EdKEUUFahhpGuokiiEeMdKLx7bTeeSBKUymaZpKQgKOhoDHFyFgrdI+aYdZXHEqNc3gNi5tReehWvZo3E1wGv4AqKNQI3a/+QOKUE2EwQ1r3fC8xQYaUoZiRvNJPNUkQHqMh6VkUiBMdZLMNcnhqnQGMpbJHGDhzf09kiGs94ZHt5MiM9GJtav5X66UmvgoyKpLUEIHnD8Upg0bCaRxwQBXBhk0sIKyo/SvEI6QQNja0ig3BX1x5GdrnNd/y/UW1flPEUQbH4AScAR9cgjq4Aw3QAhg8gmfwCt6cJ+fFeXc+5q0lp5g5BH/kfP4AkJGWTw==AAACAXicbZDLSgMxFIYz9VbrbdSN4CZYBFdlRgTdCEU3Liv0Bu0wZNJMG5rLkGSEMowbX8WNC0Xc+hbufBvTdhba+kPgy3/OITl/lDCqjed9O6WV1bX1jfJmZWt7Z3fP3T9oa5kqTFpYMqm6EdKEUUFahhpGuokiiEeMdKLx7bTeeSBKUymaZpKQgKOhoDHFyFgrdI+aYdZXHEqNc3gNi5tReehWvZo3E1wGv4AqKNQI3a/+QOKUE2EwQ1r3fC8xQYaUoZiRvNJPNUkQHqMh6VkUiBMdZLMNcnhqnQGMpbJHGDhzf09kiGs94ZHt5MiM9GJtav5X66UmvgoyKpLUEIHnD8Upg0bCaRxwQBXBhk0sIKyo/SvEI6QQNja0ig3BX1x5GdrnNd/y/UW1flPEUQbH4AScAR9cgjq4Aw3QAhg8gmfwCt6cJ+fFeXc+5q0lp5g5BH/kfP4AkJGWTw==AAACAXicbZDLSgMxFIYz9VbrbdSN4CZYBFdlRgTdCEU3Liv0Bu0wZNJMG5rLkGSEMowbX8WNC0Xc+hbufBvTdhba+kPgy3/OITl/lDCqjed9O6WV1bX1jfJmZWt7Z3fP3T9oa5kqTFpYMqm6EdKEUUFahhpGuokiiEeMdKLx7bTeeSBKUymaZpKQgKOhoDHFyFgrdI+aYdZXHEqNc3gNi5tReehWvZo3E1wGv4AqKNQI3a/+QOKUE2EwQ1r3fC8xQYaUoZiRvNJPNUkQHqMh6VkUiBMdZLMNcnhqnQGMpbJHGDhzf09kiGs94ZHt5MiM9GJtav5X66UmvgoyKpLUEIHnD8Upg0bCaRxwQBXBhk0sIKyo/SvEI6QQNja0ig3BX1x5GdrnNd/y/UW1flPEUQbH4AScAR9cgjq4Aw3QAhg8gmfwCt6cJ+fFeXc+5q0lp5g5BH/kfP4AkJGWTw==

Particle Production

REVIEW: RELAXION MECHANISM

▸ Relaxion stops at the local minima with misalignment angle close to zero- because of large e-folding number.

▸ Also, initial field velocity is zero - Contributes to CC.

▸ No DM from this !

59

‣ Classical evolution dominates over quantum fluctuation and Inflaton dominated universe.

HOW TO GET THE FIELD VELOCITY? 60

▸ Relaxion EOM:

▸ Before particle production:

▸ Dark photon begins to affect relaxion motion:

▸ After PP, relaxion motion becomes integro-differential equation difficult to solve the EOM.

FINDING ASYMPTOTIC VELOCITY

▸ Any guess to tackle the problem? ▸ Increasing velocity: Can’t be an asymptotic solution.

▸ Constant velocity: Some wave number will be produced exponentially. Always.

▸ Start with an ansatz:

▸ scale invariant dark photon production !

▸ But, the source term redshifts as, but slope potential is constant.

▸ So what’s next?

61

n = 11/a8

FINDING ASYMPTOTIC VELOCITY

‣ Find scale transformation ▸ Perturb the solution: ,

▸ Scaling behaviour using saddle point

▸ Source term scales as:

▸ Asymptotic behaviour if

▸ Asymptotic Velocity

62

n = 1� ✏AAAB+HicbZBNS8NAEIYnftb60apHL4tF8GJJRNCLUPTisYL9gDaUzXbSLt1swu5GqKG/xIsHRbz6U7z5b9y2OWjrCwsP78wws2+QCK6N6347K6tr6xubha3i9s7uXqm8f9DUcaoYNlgsYtUOqEbBJTYMNwLbiUIaBQJbweh2Wm89otI8lg9mnKAf0YHkIWfUWKtXLklyTTxy1sVEczF1Km7VnYksg5dDBXLVe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYsShqh9rPZ4RNyYp0+CWNlnzRk5v6eyGik9TgKbGdEzVAv1qbmf7VOasIrP+MySQ1KNl8UpoKYmExTIH2ukBkxtkCZ4vZWwoZUUWZsVkUbgrf45WVonlc9y/cXldpNHkcBjuAYTsGDS6jBHdShAQxSeIZXeHOenBfn3fmYt644+cwh/JHz+QP685IAAAAB+HicbZBNS8NAEIYnftb60apHL4tF8GJJRNCLUPTisYL9gDaUzXbSLt1swu5GqKG/xIsHRbz6U7z5b9y2OWjrCwsP78wws2+QCK6N6347K6tr6xubha3i9s7uXqm8f9DUcaoYNlgsYtUOqEbBJTYMNwLbiUIaBQJbweh2Wm89otI8lg9mnKAf0YHkIWfUWKtXLklyTTxy1sVEczF1Km7VnYksg5dDBXLVe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYsShqh9rPZ4RNyYp0+CWNlnzRk5v6eyGik9TgKbGdEzVAv1qbmf7VOasIrP+MySQ1KNl8UpoKYmExTIH2ukBkxtkCZ4vZWwoZUUWZsVkUbgrf45WVonlc9y/cXldpNHkcBjuAYTsGDS6jBHdShAQxSeIZXeHOenBfn3fmYt644+cwh/JHz+QP685IAAAAB+HicbZBNS8NAEIYnftb60apHL4tF8GJJRNCLUPTisYL9gDaUzXbSLt1swu5GqKG/xIsHRbz6U7z5b9y2OWjrCwsP78wws2+QCK6N6347K6tr6xubha3i9s7uXqm8f9DUcaoYNlgsYtUOqEbBJTYMNwLbiUIaBQJbweh2Wm89otI8lg9mnKAf0YHkIWfUWKtXLklyTTxy1sVEczF1Km7VnYksg5dDBXLVe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYsShqh9rPZ4RNyYp0+CWNlnzRk5v6eyGik9TgKbGdEzVAv1qbmf7VOasIrP+MySQ1KNl8UpoKYmExTIH2ukBkxtkCZ4vZWwoZUUWZsVkUbgrf45WVonlc9y/cXldpNHkcBjuAYTsGDS6jBHdShAQxSeIZXeHOenBfn3fmYt644+cwh/JHz+QP685IAAAAB+HicbZBNS8NAEIYnftb60apHL4tF8GJJRNCLUPTisYL9gDaUzXbSLt1swu5GqKG/xIsHRbz6U7z5b9y2OWjrCwsP78wws2+QCK6N6347K6tr6xubha3i9s7uXqm8f9DUcaoYNlgsYtUOqEbBJTYMNwLbiUIaBQJbweh2Wm89otI8lg9mnKAf0YHkIWfUWKtXLklyTTxy1sVEczF1Km7VnYksg5dDBXLVe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmxm2pMKBvRAXYsShqh9rPZ4RNyYp0+CWNlnzRk5v6eyGik9TgKbGdEzVAv1qbmf7VOasIrP+MySQ1KNl8UpoKYmExTIH2ukBkxtkCZ4vZWwoZUUWZsVkUbgrf45WVonlc9y/cXldpNHkcBjuAYTsGDS6jBHdShAQxSeIZXeHOenBfn3fmYt644+cwh/JHz+QP685IA

0 < ✏ ⌧ 1AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=

⌧ ! �⌧, k ! ��nkAAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=AAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=AAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=AAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=

n = 1 : ⌧ ! �⌧, k ! ��1k, ✓0! ��1✓

0

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

��8+2✏PXX(⌧PP)AAACGnicbVDNS8MwHE3n15xfVY9egkOYiKMdgjsOvXis4LbCWkuaZltYmpYkFUbp3+HFf8WLB0W8iRf/G7OPg24+CLy8934kvxemjEplWd9GaWV1bX2jvFnZ2t7Z3TP3DzoyyQQmbZywRLghkoRRTtqKKkbcVBAUh4x0w9H1xO8+ECFpwu/UOCV+jAac9ilGSkuBaUOP6XSE7vPz5lnDI6mkLOHQCXJPxNB1i5qnUDa7OU5xWgRm1apbU8BlYs9JFczhBOanFyU4iwlXmCEpe7aVKj9HQlHMSFHxMklShEdoQHqachQT6efT1Qp4opUI9hOhD1dwqv6eyFEs5TgOdTJGaigXvYn4n9fLVL/p55SnmSIczx7qZwyqBE56ghEVBCs21gRhQfVfIR4igbDSbVZ0Cfbiysuk06jbmt9eVFtX8zrK4AgcgxqwwSVogRvggDbA4BE8g1fwZjwZL8a78TGLloz5zCH4A+PrB5vIn/s=AAACGnicbVDNS8MwHE3n15xfVY9egkOYiKMdgjsOvXis4LbCWkuaZltYmpYkFUbp3+HFf8WLB0W8iRf/G7OPg24+CLy8934kvxemjEplWd9GaWV1bX2jvFnZ2t7Z3TP3DzoyyQQmbZywRLghkoRRTtqKKkbcVBAUh4x0w9H1xO8+ECFpwu/UOCV+jAac9ilGSkuBaUOP6XSE7vPz5lnDI6mkLOHQCXJPxNB1i5qnUDa7OU5xWgRm1apbU8BlYs9JFczhBOanFyU4iwlXmCEpe7aVKj9HQlHMSFHxMklShEdoQHqachQT6efT1Qp4opUI9hOhD1dwqv6eyFEs5TgOdTJGaigXvYn4n9fLVL/p55SnmSIczx7qZwyqBE56ghEVBCs21gRhQfVfIR4igbDSbVZ0Cfbiysuk06jbmt9eVFtX8zrK4AgcgxqwwSVogRvggDbA4BE8g1fwZjwZL8a78TGLloz5zCH4A+PrB5vIn/s=AAACGnicbVDNS8MwHE3n15xfVY9egkOYiKMdgjsOvXis4LbCWkuaZltYmpYkFUbp3+HFf8WLB0W8iRf/G7OPg24+CLy8934kvxemjEplWd9GaWV1bX2jvFnZ2t7Z3TP3DzoyyQQmbZywRLghkoRRTtqKKkbcVBAUh4x0w9H1xO8+ECFpwu/UOCV+jAac9ilGSkuBaUOP6XSE7vPz5lnDI6mkLOHQCXJPxNB1i5qnUDa7OU5xWgRm1apbU8BlYs9JFczhBOanFyU4iwlXmCEpe7aVKj9HQlHMSFHxMklShEdoQHqachQT6efT1Qp4opUI9hOhD1dwqv6eyFEs5TgOdTJGaigXvYn4n9fLVL/p55SnmSIczx7qZwyqBE56ghEVBCs21gRhQfVfIR4igbDSbVZ0Cfbiysuk06jbmt9eVFtX8zrK4AgcgxqwwSVogRvggDbA4BE8g1fwZjwZL8a78TGLloz5zCH4A+PrB5vIn/s=AAACGnicbVDNS8MwHE3n15xfVY9egkOYiKMdgjsOvXis4LbCWkuaZltYmpYkFUbp3+HFf8WLB0W8iRf/G7OPg24+CLy8934kvxemjEplWd9GaWV1bX2jvFnZ2t7Z3TP3DzoyyQQmbZywRLghkoRRTtqKKkbcVBAUh4x0w9H1xO8+ECFpwu/UOCV+jAac9ilGSkuBaUOP6XSE7vPz5lnDI6mkLOHQCXJPxNB1i5qnUDa7OU5xWgRm1apbU8BlYs9JFczhBOanFyU4iwlXmCEpe7aVKj9HQlHMSFHxMklShEdoQHqachQT6efT1Qp4opUI9hOhD1dwqv6eyFEs5TgOdTJGaigXvYn4n9fLVL/p55SnmSIczx7qZwyqBE56ghEVBCs21gRhQfVfIR4igbDSbVZ0Cfbiysuk06jbmt9eVFtX8zrK4AgcgxqwwSVogRvggDbA4BE8g1fwZjwZL8a78TGLloz5zCH4A+PrB5vIn/s=

NOT IMPORTANT !

‣ Find scale transformation ▸ Perturb the solution: ,

▸ Scaling behaviour using saddle point

▸ Source term scales as:

▸ Asymptotic behaviour if

▸ Asymptotic Velocity

63

n = 1� ✏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

0 < ✏ ⌧ 1AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKALF0U3LivYC3SGkknPtKGZZEgyQh36JG5cKOLWR3Hn25i2s9DWHwIf/zmHc/JHKWfaeN63U1pb39jcKm9Xdnb39qvuwWFby0xRaFHJpepGRANnAlqGGQ7dVAFJIg6daHw7q3ceQWkmxYOZpBAmZChYzCgx1uq7Ve86gFQzLkXAOfb7bs2re3PhVfALqKFCzb77FQwkzRIQhnKidc/3UhPmRBlGOUwrQaYhJXRMhtCzKEgCOsznh0/xqXUGOJbKPmHw3P09kZNE60kS2c6EmJFers3M/2q9zMRXYc5EmhkQdLEozjg2Es9SwAOmgBo+sUCoYvZWTEdEEWpsVhUbgr/85VVon9d9y/cXtcZNEUcZHaMTdIZ8dIka6A41UQtRlKFn9IrenCfnxXl3PhatJaeYOUJ/5Hz+ANLWkog=

⌧ ! �⌧, k ! ��nkAAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=AAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=AAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=AAACE3icbVDLSgMxFM3UV62vqks3wSKIaJkRQcFN0Y3LCvYBnVoymUwbmskMyR2hDP0HN/6KGxeKuHXjzr8x086iVg8ETs69h+QcLxZcg21/W4WFxaXlleJqaW19Y3OrvL3T1FGiKGvQSESq7RHNBJesARwEa8eKkdATrOUNr7N564EpzSN5B6OYdUPSlzzglICReuUjF0jiQoRdYUw+ya7H7iUezmj36Ykc42GvXLGr9gT4L3FyUkE56r3yl+tHNAmZBCqI1h3HjqGbEgWcCjYuuYlmMaFD0mcdQyUJme6mk0xjfGAUHweRMkcCnqizjpSEWo9Cz2yGBAZ6fpaJ/806CQQX3ZTLOAEm6fShIBHY5M0Kwj5XjIIYGUKo4uavmA6IIhRMjSVTgjMf+S9pnlYdw2/PKrWrvI4i2kP76BA56BzV0A2qowai6BE9o1f0Zj1ZL9a79TFdLVi5Zxf9gvX5Aw1JnZ0=

n = 1 : ⌧ ! �⌧, k ! ��1k, ✓0! ��1✓

0

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

��8+2✏PXX(⌧PP)AAACGnicbVDNS8MwHE3n15xfVY9egkOYiKMdgjsOvXis4LbCWkuaZltYmpYkFUbp3+HFf8WLB0W8iRf/G7OPg24+CLy8934kvxemjEplWd9GaWV1bX2jvFnZ2t7Z3TP3DzoyyQQmbZywRL

Documents

Abhishek Banerjee - indico.ph.tum.de · Based On: With H Kim, G. Perez (1810.01889) + D. Budkar, J. Eby (1902.08212) + O. Matsedonskii (1912.04295) + Some unpublished speculations