Dualities in and from Machine Learning
Sven Krippendorf
Deep Learning and Physics 2019
Yukawa Institute for Theoretical Physics, Kyoto
October 31st 20191

Spend 2 more slides on

Current ML applications
in high energy!2Improving sensitivity

Day, SK 1907.07642• ML-techniques heavily used in experimental bounds.

B

Cluster

AGN• Brief example: Improving sensitivity for ultra-light axion-

axion
conversion

like particles, compared to previous bounds.
• ML algorithms good at classification. Detecting particles
is a classification problem. Our classifiers:| (E)i ! ↵| (E)i + |a(E)iSpectrum

Classifier

0 (no axions)
1 (axions)galactic
absorptionγ• Training: Simulate data with and without axions forappropriate X-ray sources• Bounds: Compare fake & real data performance• Algorithms (sklearn): decision trees, boosted decision

trees, random forests, Gaussian Naive Bayes, GaussianProcess classifier, SVM, …

X-ray telescope

Previous bounds:
NGC1275: 1605.01043, Other sources: 1704.05256, Athena bounds: 1707.00176
with: Conlon, Day, Jennings;!3Berg, Muia, Powell, Rummel

NGC1275: 1605.01043
Other sources: 1704.05256
Athena bounds: 1707.00176
with: Conlon, Day, Jennings,
Rummel; Berg, Muia, Powell

Constraining ALPs

• Photon-axion interconversion in background magnetic fields:
ℒ⊃−

gaγγ
4

aFF̃ = gaγγ E ⋅ B• One interesting parameter region can be obtained forphotons from sources in and behind galaxy cluster magnetic✓◆⇣⌘ ✓ 10 3 cm 3 ◆ ✓ 1011 GeV ◆B!
fields.
?
⇥ = 0.28
Axion-photon conversion

2

⇣ p

⌘

1 µG

1 keV
ne
M
✓
◆✓
◆
⇣
⌘
ne
L
1keV
= 0.54
10 3 cm 3
10kpc
!

1 ⇥
2
2
P !a =
sin
1+⇥
Photon survival
probability
2 1 + ⇥2

sweet spot

Pγ → a0.25

no oscillations

0.20
0.15
0.10
0.05

too rapid oscillations
suppressed conversion
0.010

0.100

!4

1

10

100

ω [keV]Improving sensitivity

Picture: spectral distortui
Picture: bounds overview
Table: our results• Data: Chandra X-ray observations of bright point sources(AGN, Quasar) in or behind galaxy clusters• Bounds for ALPs with m<10-12 eV due to absence ofcharacteristic spectral modulations caused byinterconversion between photons and axions in cluster
background magnetic field
Axion-photon conversion

A1367 (resid.)
A1367 (up-resid.)
A1795 Quasar (resid.)
A1795 Quasar (up-resid.)
A1795 Sy1 (resid.)
A1795 Sy1 (up-resid.)

AB DTC GaussianNB QDA RFC Previous
C
1.9
2.4
2.0
1.0
1.1

0.8
1.1

1.9
1.7
1.2
1.1

-

-

-

1.4

-

-

1.1

0.7

1.0

0.8
!5

2.4
>10.0
>10.0
1.5
1.5

10-6
VMB
(PVLAS)

LSW (OSQAR)
10-8

Axion Coupling |GAγγ | (GeV-1)

x 10-12 GeV-1

Helioscopes (CAST)

Telescopes

Projected bounds with Athena

-10

10

Horizontal Branch Stars

SN 1987A

10-12

NGC1275 - Chandra

HESS
Fermi-LAT

NGC1275 - Athena

Haloscopes(ADMX)

-14

10

VZ

KS
DF

SZ

10-16
10-18 -30
10

10-25

10-20

10-15
10-10
Axion Mass mA (eV)

10-5

100

J Conlon et al, (1707.00176)
Francesca Day (University of Cambridge)

Axions - Motivations and Searches

LMU 201

Many talks remove slides

ML for the string landscape?
many talks: Halverson, Ruehle, Shiu

!6

Other avenues?

!7

“Don’t ask what ML can do for you,ask what you can do for ML.”– Gary Shiu

!8

Physics ⋂ ML

!9

Dualities

Betzler, SK: 191x.xxxxx!10

The problem• Obtain correlators at high(er) accuracy:

think of this as properties of your data⟨f(ϕi)⟩• In physics (incl. condensed matter, holography, string/field
theory), we often use clever data representations toevaluate correlators.

• Multiple representations can be useful ➔ dualities• Dual representations are very good representations toevaluate certain correlation functions (mapping strongly
coupled data products to weakly coupled data products).

• Examples …
!11How are dualities useful in practice?
aka connecting physics questions to data questions!12

Examples: dual representations• Discrete Fourier transformation:
1 n
xk =pj e 2πijk/n
n∑
j=1

pk =

n

∑
j=1

xj e −2πijk/n• Is there a signal under the noise?

!13

Examples: dual representations• 2D Ising model: high - low temperature self-duality
Original
H=−J

Tcritical∑⟨i, j⟩

Z=

∑

e

sisj

−βH(s)

1
β=kBT

Ordered rep.
Position space?

Dual
H=−J

σσ
∑ i j
⟨i, j⟩

Z=∑β̃ = −e −β̃H(σ)

1log tanh β2Disordered rep.

Momentum space?
Krammers,!14
Wannier 1941; Onsager 1943; review: Savit 1980

Which data problem?
• Some correlation function which is easier evaluated ondual variables.

⟨σiσj⟩, ⟨E(σ)⟩, ⟨M(σ)⟩• Can we classify the temperature for low-temperature

configurations? Which temperature is a sample drawn
from (at low temperatures)?
They look rather similar. How
about in the dual rep.?

Replace Images!15

Data question on Ising• But at the dual temperatures, our data takes a diﬀerent
shape:

Duality• It is easier to classify temperature of a low-temperature
configuration in the dual representation …

• How come?

e E/TP(s) =,Z⟨ΔE⟩ ≪ ⟨ΔẼ⟩eẼ/T̃
P(σ) =Z
ΔT ≪ ΔT̃
!16

Data question on Ising
finite• Let’s look at the overlap of energy(a)distributions
Test accuracy ofinneuralnetworks.
size samples

Original variables(a) Original temperatures.

(a) Original temperatures.

Dual variables

Let’s check for performance.

(a) Test accuracyof neural
networks.
(b) Overlap
in energy
distribution.

Figure 7: Classification of states according to their temperaturelattice Ising model. Solid lines and dots indicate data for the origina
dashed lines and crosses for the dual temperatures. Pairs of tempe
T0 + T were chosen by fixing a reference point T0 and gradually inincrements of 0.05.
!17Change figures

Ising: simple network• Let’s confirm thisat simple networks• Side remark: way
outperforming
standard sklearn
classifiers(a) Test accuracy of neural networks.

!18

Example: dual representations• 1D Ising with multiple spin interactions:normal:dual:sH(s) = − J
H(σ) = − J

N−n+1 n−1∑ ∏k=1 l=0
N−n+1

∑
k=1

σk

sk+l − BNs (here: B = 0)
∑ k
k=1

n−1

where σk = ∏ sk+l
l=0

σ

+

+

+

+

N = 10n=3

Ghost spins (fixed value)• Two data questions: 1) energy of spin configuration
2) metastable configuration!19

cf. 1605.05199

Example: dual representations• Two data questions: 1) energy of spin configuration
2) metastable configurations - - + - - - + - - +σ + + + - + + + + - +Energy++++N = 10n=3

metastable
Add evaluation of metastability on
dual and normal variables

stableconfiguration!20

cf. 1605.05199

Example: dual representations• Performance on

Normal variables

metastability classification(single hidden layer)• Deeper networks or CNNs
perform better to a certain
degree but at large N or nshow the same feature.

2
66 ·· 10102333 ·· 101033
9.5
9.5 ·· 10
103

n
n=
= 44
0.91130.9113nn== 55
0.86880.86880.92430.9243nn== 88
0.87880.87880.92150.92150.9424
0.9424nn== 99
0.88130.88130.92230.92230.94750.9475

n
n=
= 12
12
0.88030.88030.92950.92950.97390.9739(a)(a) Normal
Normal representation.
representation.

Dual variables
2
66 ·· 10102333 ·· 101033
9.5
9.5 ·· 10103(a) 600 training samples.

n
n=
= 44
0.9911
0.9911nn== 55
0.97830.97830.99580.9958nn== 88
0.98190.98190.99770.99771.00001.0000nn== 99
0.98550.98550.99940.99941.00001.0000

n
n=
= 12
12
0.99090.99091.00001.00001.00001.0000(b)(b) Dual
Dual representation.
representation.

Table
Table 1:
1: Detection
Detection ofof metastablemetastable statesstates inin thethe 1D
1D Ising
Ising chain
chain forfor di↵erentdi↵erent inte
inte
tions
and
amountsoftraining
data.
The
listednumbers
describetheaveragetions and amounts of training data. The listed numbers describe the average
test
test accuracy
accuracy over
over 1010 training
training runs
runs ofof 500500 epochs
epochs each.
each. Missing
Missing values
values indicate
indicatethethe number
number ofof required
required samples
samples exceeds
exceeds thethe total
total number
number ofof metastablemetastable statesstatestheconsideredsetting.
the considered setting.

3000
600 training
training
samplessamples
10 training runs are listed in table 1. Several examples of typical training c

10 training runs are listed in table 1. Several examples of typical training cforfor thethe casecase nn== 88 andand varying
varying amounts
amounts ofof training
training data
data cancan bebe foundfound inin figufigu

The
The results
results show
show thatthat there
there isis indeed
indeed aa major
major improvement
improvement ofof performperform
inthedual
representation.
While
all
networks
are
able
to
detectatin the dual representation. While all networks are able to detect at least
least
patternsineitherframe,
wefindseveraladvantages
from
using
the
dual
(a) 3000
600 training
samples.
!2
1samples.
(b)training
patterns
in either frame, we find several advantages from using the dual rep
reptations:

Upshot: dual representations simplify
answering certain data questions(i.e. simple networks sufficient)!22

Why interesting for ML?

!23

Why interesting for ML?
• Finding good representations which allow to answer the data questionis hard (if not impossible).

Normal Frame Dual Frame
Data Question✘

NeuralNetwork
Input Data:
Normal Frame✔

NeuralNetwork

Output: Answer
to Data Question

NeuralNetwork

Dual Representation• In this talk we use physics examples. A generalisation to other data

products/questions would be interesting. Here, the data question on
the dual network can be addressed with very simple networks.
!24

OutputDFT: simple network• Supervised learning task (binary classification):

N discrete values{((pR, pI ), y)}{((xR, xI ), y)}y = 1 noise + signal

y = 0 noise• Network
Layer

Shape

Parameters

Conv1D(2000,2)

4

Activation(2000,2)

-

Dense

1

4001

Activation

1

-

!25

For this networkclassification works inmomentum space,but not in position space.

Utilising dual representation• Goal: improve performance on position space.
• Deeper network? Can do the job in principle[DFT can be implemented with a single dense layer]• However finding it dynamically is `impossible’ with standard
optimisers, initialisations, and regularisers.
Layer
Dense
Conv1D
Activation
Dense
Activation

Shape Parameters(2000,2) 16000000(2000,2)
4(2000,2)
1
4001
1
-

DFT

!26

Randomstarting point

Can we improve the situation by
favouring dual representations?

!27

How to utilise dualities?
• Learn dual transformations when explicitly known (trivial)
and use them as intermediate step in architecture.

• Enforce dual representations via feature separation.
• When not explicitly known, we can match features of

distributions (example 2D Ising high-low-temperature
duality)• Re-obtaining “dualities” (1D Ising) by demanding good

performance on medium-hard correlation in intermediate
layer where no loss of information is present (beyond
known dualities).

!28

DFT from modified loss
• Adapt loss function to achieve feature separation, i.e. separating
the two classes of data (inspired by triplet loss)[towards generating dual representations dynamically]

Separation

Assist finding DFTwith modified loss

Loss = | ynoise |2 − | ysignal |2 + α

Decorrelation via weight regularisation

Loss = max {0, β −
+

∑
i≠j

∑

1503.03832| w |2 }DFT

max{0,(wi ⋅ wj)}

• Note: diﬀerent data question (signal injected in position andmomentum space) can lead to multiple minima in the loss
landscape, i.e. using momentum space and position space.
!29

DFT from feature separation• The final result is a simple network with few parameters
utilising `dual’ representation. (First layer indeeper network)

Network Rep.

Fourier

Input sample:!30

Feature separation in Ising• Which neural network architecture? Several architectures,
so far most promising: U-Net (1505.04597)

s-configuration Conv.

σ-configuration

Matching dual energy distribution

UpSampling
MaxPooling

Pix2Pix

UNet!31

Enforcing dual representation• Observation: deep networks do notfind these good representation bythemselves• Which ingredient is missing on theAssist finding DFTwith modified loss
DFT

neural network side?

• Work around when task which isaccessible in both frames

Add details on network• Step 1: Find better representation• Step 2: Use this representation on
previously inaccessible task
Autoencoder
Original
Original

Different rep.

fixed
!32

SophisticatedSimple Task Task

Example: 1D Ising model• Strategy from previous page shows improvement on
performance:
6 · 1023 · 103
9.5 · 102 3
6 · 103 · 103
9.5 · 103
6 · 1023 · 103
9.5 · 102 3
6 · 103 · 103
9.5 · 103

n=4

n=5

0.91130.86880.9243n=5

n=40.9113n=8n=9n = 120.87880.88130.92150.9223n=8n=90.9424
0.94750.86880.87880.88130.92150.9223(a)0.9243
Normal representation.
0.9424
0.94750.8803n0.9295= 12
0.97390.88030.92950.9739n =with4 (a)nNormal= 5 representation:n=8
n = test
9n = 12
representation.
Now“dual”
accuracy
over 99%
0.9911n=40.9911

0.97830.9958n=50.98190.98550.99770.9994n=8n=91.00001.00000.97830.98190.98550.99580.99770.9994(b)
Dual representation.
1.00001.00000.9909n1.0000= 12
1.00000.99091.00001.0000

Table 1: Detection of metastable states in the 1D Ising chain for di↵erent interactions and amounts of training
data. representation.
The listed numbers describe the average best(b) Dual!33 500 epochs each. Missing values indicate thattest accuracy over 10 training runs ofIntermediate representation• Visualise dependency of intermediate variables fi(s) andinput variables s:⟨(fi(s1, …, sj, …sN ) − fi(s1, …, − sj, …sN )) ⟩

Mij =

2

2
N
1
∑
⟨
f
(s
,
…,
s
,
…s
)
−
f
(s
,
…,
−
s
,
…s
)
k
N
i 1
k
N ) ⟩
k=1 ( i 1
N

• Resembles structure from actual dual variables (results
non-local components. The matrix Mij for one such example is presented in figure
17.

depend on initialisation), similar but not identical islands
1

1

2

2

3

3

4

4

5

5

6

6

7

7

5

4

3

2

Actual Dual variables
8

1

9
10

Intermediate variables

8
9

1

2

3

4

5

6

7

8

9

10

10

!34

1

2

3

4

5

6

7

8

9

10

Recap!35

Perspective• Large test arsenal of physics dualities from CM, holography, field/string
theory.

• Dualities are our best tools to describe strongly coupled systems.
• Allow us to get EFT-operators at higher accuracy than normally allowed(theory: large number of diagrams, experiment: large amount of data).
Think about Yukawa couplings in heterotic standard embedding• Dual data representations can be obtained via ML. We can design the

dual representation to have particular properties, e.g. display certain
quantities straight-forwardly. Can we find new useful physics dualities inthis way, i.e. new ways to describe dynamical systems? Which correlators
should we use to obtain dual representations?
• Twist to DL procedures? Intermediate layer with no information loss andrepresentation suitable for “medium-hard” task. This pre-trained structure
should be useful for more sophisticated tasks in these variables.

!36

Dualities and Fourier transformation• Review on dualities and global symmetries (Quevedo: hepth/9706210), several dualities can be seen as Fouriertransformation:“Therefore we can see that the dualities we have been dealing with for
antisymmetric tensors are only particular cases of Fourier transforms andfinding the dual action reduces to finding Fourier transforms. “• For instance duality between massive antisymmetrictensors.

Z = 𝒟B̃d−h𝒟Hh e ∫ d x(H⋅dB̃d−h+G(Hh)+F̃(B̃d−h))
∫
Hh
B̃d−h
D

S = d D x (F(∂Hh) + G(Hh))
∫

F̃

FT

e ↔ eFS̃ = d D x (F̃(B̃d−h) + G̃(∂B̃d−h))∫!37Conclusions• Dualities tightly connected to data representations.
• Finding good data representations is at the heart of ML andoften `good' representations are not found.
• `Good’ representations can be enforced when known to exist.
• This enables classification for previously `inaccessible’ tasks.
(1D Ising metastable configurations)• Interesting representations can be found from feature separation(e.g. DFT)• Interesting aspect: reduced complexity of networks• Dualities in Physics motivate multiple minima in a diﬀerent
landscape, those of the cost functions of neural networks.
• Lots of diﬀerent kinds of dualities in Physics as playground foreﬃcient neural network architectures.
!38Thank you!

!39

!40

Neural Networks• Layout of neural nets:
Weights

Parameters of network: {✓A } = {wij , bk }<latexit sha1_base64="Qwb9gPBMTS1UI6A4LY0eYZfssqU=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgQkoigm6EqhuXFewDmhAm00k7dvJg5kYpITs3/oobF4q49Rfc+TdO2yy0euDC4Zx7ufcePxFcgWV9GaW5+YXFpfJyZWV1bX3D3NxqqTiVlDVpLGLZ8YligkesCRwE6ySSkdAXrO0PL8d++45JxePoBkYJc0PSj3jAKQEteeaukzkwYEC8cyfHZ9jJ7r2M3+aH2PeGTu6ZVatmTYD/ErsgVVSg4ZmfTi+macgioIIo1bWtBNyMSOBUsLzipIolhA5Jn3U1jUjIlJtN/sjxvlZ6OIilrgjwRP05kZFQqVHo686QwEDNemPxP6+bQnDqZjxKUmARnS4KUoEhxuNQcI9LRkGMNCFUcn0rpgMiCQUdXUWHYM++/Je0jmq2VbOvj6v1iyKOMtpBe+gA2egE1dEVaqAmougBPaEX9Go8Gs/Gm/E+bS0Zxcw2+gXj4xs1/Zjf</latexit>...

Input: xi
Output: yi<latexit sha1_base64="96Omxy49SPGED3hOl6/e89unjWM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48V7Qe0oWy2k3bpZhN2N2IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0/9Tn/XLFrbpzkFXi5aQCORr98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s/mpU3JmlQEJY2VLGjJXf09kNNJ6EgW2M6JmpJe9mfif101NeOVnXCapQckWi8JUEBOT2d9kwBUyIyaWUKa4vZWwEVWUGZtOyYbgLb+8SloXVc+teneXlfp1HkcRTuAUzsGDGtThFhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP2CQjdg=</latexit>...

OutputInput<latexit sha1_base64="fYHWTPnicP3K/a8xK5ZBBgxUcng=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8eK9gPaUDbbTbt0swm7EyGE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST25nfeeLaiFg9YpZwP6IjJULBKFrpIRuIQbXm1t05yCrxClKDAs1B9as/jFkacYVMUmN6npugn1ONgkk+rfRTwxPKJnTEe5YqGnHj5/NTp+TMKkMSxtqWQjJXf0/kNDImiwLbGVEcm2VvJv7n9VIMr/1cqCRFrthiUZhKgjGZ/U2GQnOGMrOEMi3srYSNqaYMbToVG4K3/PIqaV/UPbfu3V/WGjdFHGU4gVM4Bw+uoAF30IQWMBjBM7zCmyOdF+fd+Vi0lpxi5hj+wPn8AWIWjdk=</latexit>

yi = activation (wij xj + bi )Hidden layers<latexit sha1_base64="Rx9s6iOT21Wjd0JzIm2aHR4aYjE=">AAACF3icbVDLSgNBEJyNrxhfUY9eBoMQEcKuCHoRgl48RjAPyC7L7GQ2mWT2wUxvNCz5Cy/+ihcPinjVm3/jJNmDJhY0FFXddHd5seAKTPPbyC0tr6yu5dcLG5tb2zvF3b2GihJJWZ1GIpItjygmeMjqwEGwViwZCTzBmt7geuI3h0wqHoV3MIqZE5BuyH1OCWjJLVZGLr9MbRlgQoEPp+rYFsyH8r2b8v74we2feC63Je/24NgtlsyKOQVeJFZGSihDzS1+2Z2IJgELgQqiVNsyY3BSIoFTwcYFO1EsJnRAuqytaUgCppx0+tcYH2mlg/1I6goBT9XfEykJlBoFnu4MCPTUvDcR//PaCfgXTsrDOAEW0tkiPxEYIjwJCXe4ZBTESBNCJde3YtojUkekoyzoEKz5lxdJ47RimRXr9qxUvcriyKMDdIjKyELnqIpuUA3VEUWP6Bm9ojfjyXgx3o2PWWvOyGb20R8Ynz8lY6CD</latexit>on parameters of network• Cost function dependingX
cost(✓A ) =training set|ydesired

ypredicted (✓A )|• Parameters of network updated using gradient descent<latexit sha1_base64="vBNI2e0tsdLSr0ioIglaaijYtqs=">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</latexit>✓A ! ✓A<latexit sha1_base64="Dqkmg4HtXYleum2AoErgJIvkV0w=">AAACInicbZDLSgMxFIYzXmu9VV26CRahLiwzIqi7qhuXFewFOkPJpJk2NJMZkjNCGeqruPFV3LhQ1JXgw5jpBbT1h8CX/5xDcn4/FlyDbX9ZC4tLyyurubX8+sbm1nZhZ7euo0RRVqORiFTTJ5oJLlkNOAjWjBUjoS9Yw+9fZ/XGPVOaR/IOBjHzQtKVPOCUgLHahQsXegxI+9KFCE8ZH2PXwIMriS+ye+qqENNIw7A07TlqF4p22R4Jz4MzgSKaqNoufLidiCYhk0AF0brl2DF4KVHAqWDDvJtoFhPaJ13WMihJyLSXjlYc4kPjdHAQKXMk4JH7eyIlodaD0DedIYGenq1l5n+1VgLBuZdyGSfAJB0/FCQCmziyvHCHK0ZBDAwQqrj5K6Y9oggFk2rehODMrjwP9ZOyY5ed29Ni5WoSRw7towNUQg46QxV0g6qohih6RM/oFb1ZT9aL9W59jlsXrMnMHvoj6/sHNwKjcg==</latexit>⌘ rA cost(✓A )!41

NGC1275: 1605.01043
Other sources: 1704.05256
Athena bounds: 1707.00176
with: Conlon, Day, Jennings,
Rummel; Berg, Muia, Powell

Constraining ALPs

• Photon-axion interconversion in background magnetic fields:
ℒ⊃−

gaγγ
4

aFF̃ = gaγγ E ⋅ B• One interesting parameter region can be obtained forphotons from sources in and behind galaxy cluster magnetic✓◆⇣⌘ ✓ 10 3 cm 3 ◆ ✓ 1011 GeV ◆B!
fields.
?
⇥ = 0.28
Axion-photon conversion

2

⇣ p

⌘

1 µG

1 keV
ne
M
✓
◆✓
◆
⇣
⌘
ne
L
1keV
= 0.54
10 3 cm 3
10kpc
!

1 ⇥
2
2
P !a =
sin
1+⇥
Photon survival
probability
2 1 + ⇥2

sweet spot

Pγ → a0.25

no oscillations

0.20
0.15
0.10
0.05

too rapid oscillations
suppressed conversion
0.010

0.100

!42

1

10

100

ω [keV]Standard bounds

NGC1275: 1605.01043
Other sources: 1704.05256
Athena bounds: 1707.00176
with: Conlon, Day, Jennings;
Berg, Muia, Powell, Rummel• Spectral distortions (but unknown where & strength as
dependent on magnetic field)• Poisson noise vs. signal• Essentially: fluctuations larger than noise
no realimprovement• Human-made: Fourier bounds
Projected
boundswith Athena
Axion-photon conversion

-6

10-6

VMB
VMB
(PVLAS)(PVLAS)

LSW (OSQAR)-8-1

Axion Coupling
Coupling |G|GAγγ(GeV-1))Axion
Aγγ || (GeV

10-8

Helioscopes (CAST)
-10
10

10

SN 1987A
1987A
SN

-12

10-12

Telescopes
Telescopes

Conlon, Rummel 1808.05916

Horizontal Branch
Branch Stars
Stars
Horizontal
HESS
HESS

NGC1275 -- Chandra
ChandraNGC1275

Fermi-LAT
Fermi-LAT

NGC1275 - Athena

Haloscopes(ADMX)

-14

10-14

ZZ
SVV
KKS
Z
FSSZ
DDF

-16

10-16
10

-18

10-18 -30
10
10-30
10

!43

-25

10-25
10

-20

10-20
10

-15

-10

10-15
10-10
10
10
Axion Mass
Mass mmA (eV)
(eV)
Axion
A

-5

10-5
10

0

100
10