Search for ultralight scalar dark matterwith NANOGrav pulsar timing arrays

Ryo Kato and Jiro Soda,
Institute of Cosmophysics, Kobe University.
Ryo Kato, Jiro Soda, arXiv:1904.09143.

Contents
Introduction(Review) Derivation of the detectable axion signal
in the pulsar timing array
Upper limit of the axion obtained by data analysis
SummaryPulsar

1/17

Fast spinning neutron star
Pulsar

The gravitational effects can be observed aspulse

a delay of pulse arrival times.
Earthgravitational effectPulsars can be used as a galactic-scale detector,
and sensitive to gravitational effects in a nanohertz frequency band.

Axion dark matter

2/17

Scalar field predicted by string theory
Feature: The axion behaves as a perfect fluid with an oscillating pressure.
Oscillation of pressure
Einstein's equation

Oscillation of gravitational potential
Geodesic equation

Oscillation of pulse arrival times

23
23
If the axion mass 10
is about, 22 eV
10 22
10
eV10

pulse arrival times oscillates with a nanohertz frequency.

Contents
Introduction(Review) Derivation of the detectable axion signal
in the pulsar timing array
Upper limit of the axion obtained by data analysis
SummaryFeature of the axion

3/17

To explain the energy density of the dark matter,a very large occupation number is required.
N
x3 p<latexit sha1_base64="3fVxerbA1Xks3+6M/jLcr6tEu18=">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</latexit>

⇠ 10
3

96

✓

⇢
0.4Gev/cm3

◆✓

10

23

m

eV

◆4

⇢ :the energy density of the dark matterm :the axion mass<latexit sha1_base64="EJcT7LWf5hD5Xi0ot+D32OQvnI8=">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</latexit><latexit sha1_base64="OJX0bJofVkGxZN+RbpUB20+Ul9I=">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</latexit><latexit

Axion can be treated as a classical scalar field.

The solution of the Klein-gordon equation:(t, ~x) =<latexit sha1_base64="+oiisRL+6xe94cyQdc3tb5WardM=">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</latexit>x) cos(mt + ✓(~x))
0 (~becausek ⇠ (10kpc) 1 ⇠ 105 H0

dispersion relation:<latexit sha1_base64="eicdUwQjkdX6kd7vJnouOknVXOE=">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</latexit>

Galaxy Scale

Ek2 = k 2 + m2 ' m2
(monochromatic with m)<latexit sha1_base64="3aad741XOMqqOsEEHplGqM8Mdww=">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</latexit>m ⇠ 10 23 eV ⇠ 1010 H0<latexit sha1_base64="oXpyBsZTOk/RO0ss/z71ny5aQFc=">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</latexit>

Energy momentum tensor
Tµ⌫ = @µ @⌫<latexit sha1_base64="iOGAvXIsWnI+mK9zbhqsqJV2hyM=">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</latexit>

4/171((@ )2 + m2 )2

Using the equation:(t, ~x) =x) cos(mt + ✓(~x))
0 (~with @i⌧ @0<latexit sha1_base64="+oiisRL+6xe94cyQdc3tb5WardM=">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</latexit><latexit sha1_base64="r5E8Z1kDmqKjYxlMGEHd86saTcg=">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</latexit>*k⌧m<latexit sha1_base64="/JjbPusKMV/yE2WdGnWaP7IDYNM=">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</latexit>

1 ˙2 1 2 2
1 2 2
T00 =
+ m
= m 0
2
2
2
✓
◆
1 ˙2 1 2 2
1 2 2
Tij =
m
ij = m 0 cos(2mt + ✓) ij2
2
2<latexit sha1_base64="hok0n8HsfqFdT2OKmCHH+Ophh60=">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</latexit><latexit sha1_base64="CCFQiydQWNUE0PHxIyr4AkoBHT8=">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</latexit>

The axion behaves as a perfect fluid with an oscillating pressure.

Einstein's equation
Newtonian gaugeGµ⌫ = 8⇡GTµ⌫ds2 =<latexit sha1_base64="ObuDu9a2+yJwfj6vWm1GJ/ZN25c=">AAACgHichVG7SgNBFD2u7/iK2gg2wRCxindFUARBtNDSV0wgkbC7jjpkX+wjoMHC1h+wsFIQERv9Bht/wMJPEEsFGwtvNouiot5hZs6cuefOmRndNaUfED00Kc0trW3tHZ2Jru6e3r5k/8CG74SeIXKGYzpeQdd8YUpb5AIZmKLgekKzdFPk9cpCfT9fFZ4vHXs92HPFpqXt2HJbGlrAVDk5vFiulaywZIcHs9MlV6YW1z+IcjJNWYoi9ROoMUgjjmUneYEStuDAQAgLAjYCxiY0+NyKUEFwmdtEjTmPkYz2BQ6QYG3IWYIzNGYrPO7wqhizNq/rNf1IbfApJnePlSlk6J4u6Znu6Ioe6e3XWrWoRt3LHs96Qyvcct/R0NrrvyqL5wC7n6o/PQfYxnTkVbJ3N2LqtzAa+ur+8fPazGqmNkpn9MT+T+mBbvkGdvXFOF8RqydI8Aeo35/7J9iYyKqUVVcm03Pz8Vd0YBgjGOP3nsIclrCMHJ97iAtc40ZRlDFlXFEbqUpTrBnEl1Bm3gFjMJQK</latexit>
sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">AAACVXichVG7SgNBFD1ZX3F9JMFGsAmGiFW4a6NYCTaWeZgHxBB2N6Mu2ewuu5tADP5AWgsLKwUR8TNs/AGL9DZiGcHGwrubgKiod5iZM2fm3DlzR3NMw/OJBhFpYnJqeiY6K8/NywuLsfh8ybPbri6Kum3abkVTPWEalij6hm+KiuMKtaWZoqw1d4P9cke4nmFb+37XEbWWemQZh4au+kxl6/EUZSiM5E+gjEEK47DjNzhAAzZ0tNGCgAWfsQkVHrcqFBAc5mroMecyMsJ9gVPIrG3zKcEnVGabPB7xqjpmLV4HOb1QrfMtJneXlUmk6ZFuaUgPdEfP9P5rrl6YI/DS5VkbaYVTj/WXC2//qlo8+zj+VP3p2cchtkKvBnt3QiZ4hT7Sd07Oh4XtfLq3Rlf0wv4vaUD3/AKr86pf50T+AjLXX/le7Z+gtJFRKKPkCFGsYBXrXOZN7GAPWRT5ugb6OIs8SbKUGP2TFBl/WAJfQlr6AAYJh7c=</latexit>
sha1_base64="srvHxwNRKiamW53Qoh7euRMnGto=">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</latexit>
sha1_base64="iSVV4VniVbWAm838nicPwdgRveM=">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</latexit>(1 + 2 )dt2 + (1 + 2 ) ij dxi dxj<latexit sha1_base64="TgIg0jYLeQDhsSx00tEaT38a8n0=">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</latexit>=

00 component:<latexit sha1_base64="B0sfjYtaDmBwe75rATYDb/KwGPo=">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</latexit>

1 2 2⇢⌘ m 02

4⇡G⇢¨ = 4⇡G⇢ cos(2mt + ✓)ij component:<latexit sha1_base64="9VfDFBi74iiHGmDWtFSK1FBG2lQ=">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</latexit><latexit sha1_base64="79Qz74idfUTnfLOD7RuOwFd/v5U=">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</latexit>

Separate<latexit sha1_base64="VGpuaGUbT8klfcnCLo43U9p2tE0=">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</latexit>

into time-independent and time-dependent terms.

=<latexit sha1_base64="GLBA2ALHxd0SF6yiPJ2SAAVnAyo=">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</latexit>

Then<latexit sha1_base64="cfsCWgc6AW2chmVUxDXv0N5VgD4=">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</latexit>

0 =

x) +
0 (~

(t, ~x)¨ = 4⇡G⇢ cos(2mt + ✓)4⇡G⇢<latexit sha1_base64="0pyuldZLq4lonUwBCObTiNu92PQ=">AAACknichVHLShxBFD22xsck0fGxENwMDoohONwWIUEQNC504WLUjAq2DN09pVPY3dV01wzo4A+4deHCVYQQxC/INm78ARd+grg04MZF7vQ0hCiaW1TVqVP33DpV5YSejDXRTZvR3vGms6u7J/P23fvevmz/wHqsapErSq7yVLTp2LHwZCBKWmpPbIaRsH3HExvO3kJzf6Muoliq4KveD8W2b+8Gcke6tmaqnJ20KhWlG1ZFeNq2irE8nJ22QplbzFlRVVmuiiemfP3R0lWh7Q/lbJ4KlETuOTBTkEcaRZX9AQsVKLiowYdAAM3Yg42Y2xZMEELmttFgLmIkk32BQ2RYW+MswRk2s3s87vJqK2UDXjdrxona5VM87hErcxijazqne7qiC7qlxxdrNZIaTS/7PDstrQjLfUfDaw//Vfk8a1T/ql71rLGDz4lXyd7DhGnewm3p6wcn92szq2ONcTqjO/b/jW7okm8Q1H+731fE6iky/AHm0+d+DtanCiYVzJXp/NyX9Cu6MYJRTPB7f8IcllBEic89xk/8wqUxZMwY88ZCK9VoSzWD+CeM5T8WBprL</latexit>

Undetectable= ⇡G⇢ cos(2mt + ✓)<latexit sha1_base64="qsNj3iUILDt/v/bwLrMnIyVKy50=">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</latexit>

Oscillation of the gravitational potential
is induced from the pressure oscillation.
<latexit sha1_base64="p3BUgcx66CMr/9sDcKTKnE8c1cE=">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</latexit>

5/17

Geodesic equation

6/17

To obtain the red shift of the light,
the zero component of the geodetic equation is calculated.

<latexit sha1_base64="vLwKWEl6OOpFvhJBnrfi+jUue8s=">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</latexit>

d2 x0
+
d

⌫
⇢
dx
dx
0
=0
⌫⇢
d d

<latexit sha1_base64="VKR3AXNcf8huRJYRS6nIX6Nec0I=">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</latexit>

d2 x0
=
d

ds2 =

(1 + 2)dt2 + (1 + 2<latexit sha1_base64="1nOX7faM6E2uAZL8mE+DQPPJnZ0=">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</latexit>

d
2!0
d

) ij dxi dxj!0 :unperturbed frequency<latexit sha1_base64="MzrX57R6aq24e3U9W+6ZcdpMFT4=">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</latexit>

Observed light frequency:
Four-velocity at Earth and pulsar!obs = !0<latexit sha1_base64="egmLrdVVmppirQRqAMdA7ar+ATk=">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</latexit>!0 [= !0 + !0 [( Pulsar )( Pulsar )( Earth )] + 2!0( Earth )]Z

Pulsar

Earth

2 0
d
x
0
d
d

The delay of the light
Red shift:!0z=<latexit sha1_base64="i3TD40bMkOzRJaaZ6f/cUtn+wmM=">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</latexit>

!obs

=

( Earth )!0= ⇡G[⇢(~xE ) cos(2mt + ✓(~xE ))7/17( Pulsar )⇢(~xP ) cos(2m(tD) + ✓(~xP ))]⇢(~xP ) sin(2m(tD) + ✓(~xP ))]The delay of the light:
Z t
t=
dt z
0

<latexit sha1_base64="RcQ4F7hvXYKYfdLzWp4dE0n3j+g=">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</latexit>⇡G=[⇢(~xE ) sin(2mt + ✓(~xE ))2m

The delay of the light t can be detected by the pulser timing arrayas a change in the pulse arrival times.
<latexit sha1_base64="BnjNMdV5GbcdwEIx9vwm1jvgI0Y=">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</latexit>

Contents
Introduction(Review) Derivation of the detectable axion signal
in the pulsar timing array
Upper limit of the axion obtained by data analysis
SummaryAxion signal
The delay of the pulse arrival time1
[ sin(2⇡f t + ↵)2⇡f<latexit sha1_base64="IN2qvrep9tgpnFzgHJ4fgCaBRJY=">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</latexit>

⇡G⇢
⌘
m2

m : Axion mass,sin(2⇡f t + ↵p )]mf⌘⇡⇢ : Axion energy density↵, ↵p : Phases at earth and pulsar
In this study:
We search for the axion, and if it cannot be detected,
we give an upper limit for .
We investigate whether the axion is necessary
for a model which describes the observation data.

8/17

Bayesian statistics
Parameter estimation
Bayes' theorem:~ / p(d|✓)p(~ ✓)~p(✓|d)✓~ : parameterd : observation data<latexit sha1_base64="vAAQiIW1TEU0G0b5qR6TU+Uo+9o=">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</latexit>

The goal of the Bayesian estimation is to update~
the prior knowledge of the parameter p(✓)~
to the posterior knowledge p(✓|d)
using the data d .:<latexit sha1_base64="O/DxLmTM+qCt0IMG364qhwRtuTE=">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</latexit><latexit sha1_base64="gWvB3Pvb88+WJq27EghYwUnz2pM=">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</latexit>

Bayesian hypothesis testing
Bayes factor:p(d|M1 )p( |M1 )B10 ⌘= M0 , M1 : hypothesisp(d|M0 )p( |d, M1 ) = ⇤<latexit sha1_base64="Lf2uSNfj4qv/ON5xpIybWm/JfmY=">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</latexit>

The goal of the Bayesian hypothesis testing is
to compute the Bayes factor B10 .
<latexit sha1_base64="txMN4o5+7seSv4HDxoEHZXKiUxc=">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</latexit>

Interpretation of the Bayes factor(Jeffreys’ scale)

B10
1-3
3-20
20-150

Evidence in favor of M0

Week
Positive
Strong

9/17

Data analysis

10/17

Pulsar observation data: The NANOGrav Eleven-Year Data Set
https://data.nanograv.org

Program: Based on PAL2 (Bayesian inference package for pulsar timing data)
https://github.com/jellis18/PAL2

First Analysis
Model: s + nred<latexit sha1_base64="yrULF+u9DJ3O9s8EO2wQv0oACsc=">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</latexit>s : Axion signal<latexit sha1_base64="mwLspu+jiueYgCR7Y+AUHZfiogw=">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</latexit><latexit
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sha1_base64="5mCdhMx05kpYcEUVeJMuz8S6xCU=">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</latexit>
sha1_base64="NGB2mgxraatRLbRzWfETykK/Kwc=">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</latexit>

nred : Red noise<latexit sha1_base64="64fNPeZVXXakXWMNzKmuXpnv3Ds=">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</latexit>

Red noise: The power spectral density of the noise has most of their power
at low frequencies in a given data set.
Source: ・Fluctuations of the rotation speed of the pulsar
・Diffractive and refractive interstellar effects
Z 1
hnred nTred i =df e 2⇡if t P (f )P (f ) ⌘ Af1

First analysis
Upper limit on the axion amplitude

11/17

as a function of the frequency

sensitive frequency band<latexit
sha1_base64="pkNZZX0VTk2LY78WGBha28D77/U=">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</latexit>⇢=0.4 GeVcm

3

3-20: positive evidence

1-3: weak evidence

1/11yr

The upper limit of the amplitude is still one order of magnitude larger than the expected amplitude.
The Bayes factor exceeds 20, and so the existence of the axion can not be denied.

Uncertainty of earth orbit

12/17

In the case of gravitational waves
Uncertainties of the earth orbit affects
upper limits and Bayes factors.
Z. Arzoumanian et al. The NANOGrav 11-year Data Set: Pulsar-timing Constraints On The Stochastic Gravitational-wave Background. Astrophys. J., 859(1):47, 2018.

Second Analysis
Model: s + nred + norbit<latexit sha1_base64="8bIxp7bGktFODvjyyfAlKZWH5QU=">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</latexit>s : Axion signal<latexit sha1_base64="mwLspu+jiueYgCR7Y+AUHZfiogw=">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</latexit><latexit
sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit>
sha1_base64="5mCdhMx05kpYcEUVeJMuz8S6xCU=">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</latexit>
sha1_base64="NGB2mgxraatRLbRzWfETykK/Kwc=">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</latexit>

nred : Red noise<latexit sha1_base64="64fNPeZVXXakXWMNzKmuXpnv3Ds=">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</latexit>

norbit : Uncertainties of earth orbit<latexit sha1_base64="JdxZlbRNaaU7vuJNDuo6iqyMdEw=">AAACcHichVHLLgRBFD3TXmM8ZrCRWBgmRCwmt0VCrCZsLL0GCZNJdytU9Et3zSRM/IAfsLBBIiI+w8YPWPgEsUNiY+FOTyeC4Faq6tSpe26dqjJ9W4aK6CGhNTW3tLYl21MdnV3d6UxP72roVQJLFC3P9oJ10wiFLV1RVFLZYt0PhOGYtlgz9+bq+2tVEYTSc1fUgS9KjrHjym1pGYqpkluubQZO1gtMqY7KmRzlKYrsT6DHIIc4FrzMFTaxBQ8WKnAg4EIxtmEg5LYBHQSfuRJqzAWMZLQvcIQUayucJTjDYHaPxx1ebcSsy+t6zTBSW3yKzT1gZRYjdE/X9EJ3dEOP9P5rrVpUo+7lgGezoRV+OX3cv/z2r8rhWWH3U/WnZ4VtTEdeJXv3I6Z+C6uhrx6evCzPLI3URumCntj/OT3QLd/Arb5al4ti6RQp/gD9+3P/BKsTeZ3y+uJkrjAbf0USAxjGGL/3FAqYxwKKfO4+TnCG88Sz1q8NakONVC0Ra/rwJbTxD2b1jzg=</latexit>

Sources of norbit :<latexit sha1_base64="JdxZlbRNaaU7vuJNDuo6iqyMdEw=">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</latexit>

・ Error of ecliptic longitude
・ Mass errors of Jupiter, Saturn, Uranus, and Neptune
・ Orbit error of Jupiter
The Jupiter has a orbital periodof 11.86 years and a large mass.

The Jupiter produces a large noisein the low frequency regionof pulser timing arrays.

Second analysis
Upper limit on the axion amplitude

as a function of the frequency

sensitive frequency band<latexit
sha1_base64="pkNZZX0VTk2LY78WGBha28D77/U=">AAAChHichVHLSuRAFD3Gd4/arW4EF4ZpHFy0zY0PFEGRcaFLX90KRpsklhrMiyTdoKGXbvwBF65GkGFwq1/gxh9w4SeISwU3LrydDgwq6i2q6tSpe26dqtI9ywxCorsGqbGpuaW1rT31o6OzK53p7ikGbtk3RMFwLddf17VAWKYjCqEZWmLd84Vm65ZY0/fnavtrFeEHpuushgee2LS1XcfcMQ0tZKqUGVD9PVeelik/pubkSPVteV4U1Zxhb0XDo9VqKZOlPMUhfwRKArJIYtHN/IWKbbgwUIYNAQchYwsaAm4bUEDwmNtExJzPyIz3BapIsbbMWYIzNGb3edzl1UbCOryu1QxitcGnWNx9VsoYpFv6R490Qxd0Ty+f1oriGjUvBzzrda3wSunjvpXnb1U2zyH2/qu+9BxiB5OxV5O9ezFTu4VR11cOTx5XppYHo190Rg/s/w/d0TXfwKk8GedLYvkUKf4A5f1zfwTFkbxCeWVpLDv7O/mKNvTjJ4b4vScwiwUsosDnHuECl7iSWqScNCqN11OlhkTTizchzbwC1qSTnQ==</latexit>⇢=0.4 GeV

cm

13/17

3

3-20: positive evidence

1-3: weak evidence

1/11yrAlmost the same upper limit as in the first analysis is obtained,
but most of the Bayes factors are less than 3, and so no axion is detected.

Comparison with previous study 14/17
Upper limit on the axion amplitude

as a function of the frequency

PPTA2018

sensitive frequency band<latexit
sha1_base64="pkNZZX0VTk2LY78WGBha28D77/U=">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</latexit>⇢=0.4 GeVcm

3

3-20: positive evidence

1-3: weak evidence

1/11yrPrevious study: Parkes Pulsar Timing Array Twelve-Year Data Set (PPTA2018)
N. K. Porayko et al ., Phys. Rev. D 98 (2018) no.10, 102002<latexit sha1_base64="HM1MZarY+S0q05xXY/YCUEyUdFY=">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</latexit>

Up to three times stronger upper limit is obtained in the blue circle.

Fixed noise analysis,

Bayes factor isless than 3<latexit sha1_base64="14KZWP9mKfEbWCBO6w2co28E8Is=">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</latexit><latexit

15/17

Probability that axion is not required ismore than 75%

We would like to know whether the noise
is sufficient to absorb the signal of the axion.
Last Analysiss + nred + norbit
Model:<latexit sha1_base64="8bIxp7bGktFODvjyyfAlKZWH5QU=">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</latexit>s : Axion signal<latexit sha1_base64="mwLspu+jiueYgCR7Y+AUHZfiogw=">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</latexit><latexit
sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit>
sha1_base64="5mCdhMx05kpYcEUVeJMuz8S6xCU=">AAACWXichVG7SgNBFD1ZXzFGExVBsAmGiFW4a6NYCTaWeZgHxBB210lc3Owuu5tADP6AtoqFlYKI+Bk2/oBFfkAQywg2Ft5sAqJBvcPMnDkz586Zuapt6K5H1AlII6Nj4xPBydBUeHomEp0N512r4Wgip1mG5RRVxRWGboqcp3uGKNqOUOqqIQrq4XZvv9AUjqtb5q7XskW5rtRMvaprisdU2q1E45QkP2LDQB6AOAaRsqK32MM+LGhooA4BEx5jAwpcbiXIINjMldFmzmGk+/sCxwixtsGnBJ9QmD3kscar0oA1ed3L6fpqjW8xuDusjCFBT3RHXXqke3qhj19ztf0cPS8tntW+VtiVyMli9v1fVZ1nDwdfqj89e6hiw/eqs3fbZ3qv0Pr65tFFN7uZSbRX6Jpe2f8VdeiBX2A237SbtMhcIsQFkH9+9zDIryVlSsppQhBLWMYqf/M6trCDFHJ8ncApznAeeJbC0ny/VFJgULM5fAtp4RP254kj</latexit>
sha1_base64="NGB2mgxraatRLbRzWfETykK/Kwc=">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</latexit>

thenModel: s + nred + norbit<latexit sha1_base64="8bIxp7bGktFODvjyyfAlKZWH5QU=">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</latexit>

nred : Red noise<latexit sha1_base64="64fNPeZVXXakXWMNzKmuXpnv3Ds=">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</latexit>

norbit : Uncertainty of earth’s orbit<latexit sha1_base64="JdxZlbRNaaU7vuJNDuo6iqyMdEw=">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</latexit>

Note that,
It is difficult to distinguish the signal and the noise.
Therefore, this analysis cannot be regarded as
giving upper limits to the axion.

Last analysis
Upper limit on the axion amplitude

16/17

as a function of the frequency

sensitive frequency band<latexit
sha1_base64="pkNZZX0VTk2LY78WGBha28D77/U=">AAAChHichVHLSuRAFD3Gd4/arW4EF4ZpHFy0zY0PFEGRcaFLX90KRpsklhrMiyTdoKGXbvwBF65GkGFwq1/gxh9w4SeISwU3LrydDgwq6i2q6tSpe26dqtI9ywxCorsGqbGpuaW1rT31o6OzK53p7ikGbtk3RMFwLddf17VAWKYjCqEZWmLd84Vm65ZY0/fnavtrFeEHpuushgee2LS1XcfcMQ0tZKqUGVD9PVeelik/pubkSPVteV4U1Zxhb0XDo9VqKZOlPMUhfwRKArJIYtHN/IWKbbgwUIYNAQchYwsaAm4bUEDwmNtExJzPyIz3BapIsbbMWYIzNGb3edzl1UbCOryu1QxitcGnWNx9VsoYpFv6R490Qxd0Ty+f1oriGjUvBzzrda3wSunjvpXnb1U2zyH2/qu+9BxiB5OxV5O9ezFTu4VR11cOTx5XppYHo190Rg/s/w/d0TXfwKk8GedLYvkUKf4A5f1zfwTFkbxCeWVpLDv7O/mKNvTjJ4b4vScwiwUsosDnHuECl7iSWqScNCqN11OlhkTTizchzbwC1qSTnQ==</latexit>⇢=0.4 GeVcm

3<1: no evidence

1/11yr

The model of only nred and norbit absorbs the axion signal sufficiently.
It suggests that the axion is unnecessary for the model of the observation data.
<latexit sha1_base64="64fNPeZVXXakXWMNzKmuXpnv3Ds=">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</latexit><latexit sha1_base64="JdxZlbRNaaU7vuJNDuo6iqyMdEw=">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</latexit>

Summary

17/17

We set the upper limit of the axion amplitude
using the recent pulsar observation data.
By considering the uncertainty of the earth’s orbit,
we revealed that the axion is not detected.

In comparison with the previous study (Porayko et.al.2018),
we obtain up to three times stronger upper limit.

We clarified that the only the red noise andthe uncertainty of the earth’s orbit
are sufficient for the model of observation data.