Post on 16-Jan-2017
Finding self-force quantities in a post-Newtonian expansion
Seth Hopper
Eccentric orbits on a Schwarzschild background
π�
π�π�
�π
-�
-�
-�
-�
-�
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
Chris Kavanagh
Adrian Ottewill
Erik Forseth
Charles Evans
Institut des Hautes Études Scientifiques - November 12, 2015
An electron in free fall will radiate
e-
e-
e-
An electron is a non-local object
Outline
Orbits
Homogeneous solutions
Particular solutions
Black Hole Perturbation Theory
Formalism
Analytic PN solutions
Numeric PN solutions
Outline
Orbits
Homogeneous solutions
Particular solutions
Black Hole Perturbation Theory
Formalism
Analytic PN solutions
Numeric PN solutions
In black hole perturbation theory, we alternate
between correcting fields and particle motion
Field equations Equations of motion
0
1
2
Order
G
µ⌫
= 0
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
G
µ⌫
= 0
Du
µ
d⌧
= 0
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
g
µ⌫
= g
BHµ⌫
+ p
µ⌫
2pµ⌫
+ 2R
↵µ�⌫
p
↵�
= �16⇡T
µ⌫
G
µ⌫
= 0
Du
µ
d⌧
= 0
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
g
µ⌫
= g
BHµ⌫
+ p
µ⌫
2pµ⌫
+ 2R
↵µ�⌫
p
↵�
= �16⇡T
µ⌫
G
µ⌫
= 0
Du
µ
d⌧
= 0
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
Du
µ
d⌧
= F
µ
self [p↵� ]
Du
µ
d⌧
= 0
g
µ⌫
= g
BHµ⌫
+ p
µ⌫
2pµ⌫
+ 2R
↵µ�⌫
p
↵�
= �16⇡T
µ⌫
G
µ⌫
= 0
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
2p2 = (1 + p1)T [z + �z] + (rp1)2
Du
µ
d⌧
= F
µ
self [p↵� ]
Du
µ
d⌧
= 0
g
µ⌫
= g
BHµ⌫
+ p
µ⌫
2pµ⌫
+ 2R
↵µ�⌫
p
↵�
= �16⇡T
µ⌫
G
µ⌫
= 0
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
Eccentric orbits on Schwarzschild have two
frequencies
Tr
'(t)
'(Tr)
�'(t)
t
0 t Tr
rmin ! rmin
1
Tr
'(t)
'(Tr)
�'(t)
t
0 t Tr
rmin ! rmin
1
⌦
r
=
2⇡
T
r
⌦
'
=
'
p
(T
r
)
T
r
�'(t) = '
p
(t)� ⌦
'
t
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`!
(r) = 0
f
d
dr
✓f
d
dr
◆+ !
2 � V
l
(r)
�X
lm!
(r) = 0
7
Tr
1
Tr
'(t)
t
1
x =
�M⌦
'
�2/3
T
r
'
p
(t)
'(T
r
)
�'(t)
X
1
=
M
r
X
2
= (!r)
2
4
x =
�M⌦
'
�2/3
T
r
'
p
(t)
'
p
(T
r
)
�'(t)
X
1
=
M
r
X
2
= (!r)
2
4
The radial frequency is oscillatory, but the azimuthal
frequency is not
Tr
1
Tr
'(t)
t
1
Tr
'(t)
'(Tr)
�'(t)
t
1
⌦
r
=
2⇡
T
r
⌦
'
=
'(T
r
)
T
r
�'(t) = '
p
(t)� ⌦
'
t
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`!
(r) = 0
f
d
dr
✓f
d
dr
◆+ !
2 � V
l
(r)
�X
lm!
(r) = 0
7
Tr
1
Tr
'(t)
t
1
x =
�M⌦
'
�2/3
T
r
'
p
(t)
'(T
r
)
�'(t)
X
1
=
M
r
X
2
= (!r)
2
4
x =
�M⌦
'
�2/3
T
r
'
p
(t)
'
p
(T
r
)
�'(t)
X
1
=
M
r
X
2
= (!r)
2
4
⌦
r
=
2⇡
T
r
⌦
'
=
'
p
(T
r
)
T
r
�'(t) = '
p
(t)� ⌦
'
t
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`!
(r) = 0
f
d
dr
✓f
d
dr
◆+ !
2 � V
l
(r)
�X
lm!
(r) = 0
7
Darwin’s parametrization puts the orbit into a
“Newtonian-like” form
⌦
'
=
1
M
1� e
2
p
!3/2
"1 + 3
e
2
p
+O⇣p
�2
⌘#
r
p
(�) =
pM
1 + e cos�
x =
�M⌦
'
�2/3
T
r
'(t)
'(T
r
)
�'(t)
1
pM
C
±=
1
WT
r
ZT
r
0
"1
f
p
X
⌥G(t)+
2M
r
2
p
f
2
p
X
⌥ � 1
f
p
dX
⌥dr
!F (t)
#e
i!t
dt
'
p
(�)
C
±=
1
WT
r
Z2⇡
0
"1
f
p
X
⌥¯
G+
2M
r
2
p
f
2
p
X
⌥ � 1
f
p
dX
⌥dr
!¯
F
#e
in⌦
r
t
dt
p
d�
d�
�'(�)
r
p
(�) =
hM �M cos(�)e +O
⇣e
2
⌘ ix
�1
+
h2e
2
M � 2e
3
M cos(�) +O⇣e
4
⌘ ix
0
+O⇣x
1
⌘.
(p, e) = (10, 0.5)
0 � 2⇡ () 0 t T
r
pM
C
±=
1
WT
r
ZT
r
0
"1
f
p
X
⌥G(t)+
2M
r
2
p
f
2
p
X
⌥ � 1
f
p
dX
⌥dr
!F (t)
#e
i!t
dt
'
p
(�)
C
±=
1
WT
r
Z2⇡
0
"1
f
p
X
⌥¯
G+
2M
r
2
p
f
2
p
X
⌥ � 1
f
p
dX
⌥dr
!¯
F
#e
in⌦
r
t
dt
p
d�
d�
�'(�)
r
p
(�) =
hM �M cos(�)e +O
⇣e
2
⌘ ix
�1
+
h2e
2
M � 2e
3
M cos(�) +O⇣e
4
⌘ ix
0
+O⇣x
1
⌘.
dt
p
d�
=
p
2
M
(p� 2� 2e cos�)(1 + e cos�)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
#1/2
'
p
(�) =
✓4p
p� 6� 2e
◆1/2
F
✓�
2
�����4e
p� 6� 2e
◆
⌦
'
=
1
M
1� e
2
p
!3/2
"1 + 3
e
2
p
+O⇣p
�2
⌘#
r
p
(�) =
pM
1 + e cos�
1
t
p
(�) =
Z�
0
p
2
M
(p� 2� 2e cos�
0)(1 + e cos�
0)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
0
#1/2
d�
0
One radial period
The metric perturbation in Regge-Wheeler gauge is
found by solving a wave equation
"� @
2
@t
2
+
@
2
@r
2⇤� V
`
(r)
#
`m
(t, r) = S
`m
(t, r)
�'(�) = '
p
(�)� ⌦'
t
p
(�)
'
p
(�) = �+
h3� + sin(�)e + 3�e
2
+O⇣e
3
⌘ix+O
⇣x
2
⌘
�'(�) =
2e sin(�)� 3
4
e
2
sin(2�) +O⇣e
3
⌘�x
0
+
4e sin(�)� 3
4
e
2
sin(2�) +O⇣e
3
⌘�x
1
+O⇣x
2
⌘
TD master equation
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`mn
(r) = Z
`mn
(r)
! ⌘ !
mn
= m⌦
'
+ n⌦
r
`m
(t, r) =
1X
n=�1R
`mn
(r) e
�i!t
S
`m
(t, r) =
1X
n=�1Z
`mn
(r) e
�i!t
FD master equation
`m
(t, r) =
1X
n=�1R
`mn
(r) e
�i!t
S
`m
(t, r) =
1X
n=�1Z
`mn
(r) e
�i!t
"� @
2
@t
2
+
@
2
@r
2⇤� V
`
(r)
#
`m
(t, r) = S
`m
(t, r)
�'(�) = '
p
(�)� ⌦'
t
p
(�)
'
p
(�) = �+
h3� + sin(�)e + 3�e
2
+O⇣e
3
⌘ix+O
⇣x
2
⌘
! ⌘ !
mn
= m⌦
'
+ n⌦
r
`m
(t, r) =
1X
n=�1R
`mn
(r) e
�i!t
S
`m
(t, r) =
1X
n=�1Z
`mn
(r) e
�i!t
"� @
2
@t
2
+
@
2
@r
2⇤� V
`
(r)
#
`m
(t, r) = S
`m
(t, r)
�'(�) = '
p
(�)� ⌦'
t
p
(�)
Fourier Series
Doubly periodic
Exponential convergence is achieved through
extended homogeneous solutions
10-13
10-11
10-9
10-7
10-5
10-3
10-1
3 4 5 6 7 8 9 10 11 12 13 14
r/M
log
!
|Ψstd
22
(nm
ax)−
Ψstd
22
(N)|"
rmin rmax
N = 3
N = 5
N = 8
N = 10
N = 12
N = 14
10-13
10-11
10-9
10-7
10-5
10-3
10-1
3 4 5 6 7 8 9 10 11 12 13 14
r/M
log
!
|ΨE
HS
22
(nm
ax)−
ΨE
HS
22
(N)|"
rmin rmax
N = 3
N = 5
N = 8
N = 10
N = 12
N = 14
R
±`mn
(r) ⌘ C
±`mn
ˆ
R
±`mn
(r)
±`m
(t, r) =
X
n
C
±`mn
ˆ
R
±`mn
(r) e
�i!t
r > 2M
`m
(t, r) =
+
`m
(t, r)✓
⇥r � r
p
(t)
⇤+
�`m
(t, r)✓
⇥r
p
(t)� r
⇤
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`mn
(r) = Z
`mn
(r)
! ⌘ !
mn
= m⌦
'
+ n⌦
r
R
±`mn
(r) ⌘ C
±`mn
ˆ
R
±`mn
(r)
±`m
(t, r) =
X
n
C
±`mn
ˆ
R
±`mn
(r) e
�i!t
r > 2M
`m
(t, r) =
+
`m
(t, r)✓
⇥r � r
p
(t)
⇤+
�`m
(t, r)✓
⇥r
p
(t)� r
⇤
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`mn
(r) = Z
`mn
(r)
! ⌘ !
mn
= m⌦
'
+ n⌦
r
Time domain EHS
Normalization coefficients are found by integrating a
radially-periodic function
C
±`mn
=
1
W
`mn
T
r
ZT
r
0
"1
f
p
ˆ
R
⌥`mn
(r
p
)G
`m
(t)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(r
p
)� 1
f
p
d
ˆ
R
⌥`mn
(r
p
)
dr
!F
`m
(t)
#e
i!t
dt
'
p
(�)
C
±=
1
WT
r
Z 2⇡
0
"1
f
p
X
⌥¯
G(t) +
✓2M
r
2p
f
2p
X
⌥ � 1
f
p
dX
⌥
dr
◆¯
F (t)
#e
in⌦r
t
dt
p
d�
d�
�'(�)
r
p
(�) =
hM � eM cos(�) +O �e2�
ix
�1
+
h2e
2M � 2e
3M cos(�) +O �e4�
ix
0+O �x1� .
¯
G
`m
(t) ⌘ G
`m
(t) e
im⌦
'
t
,
¯
F
`m
(t) ⌘ F
`m
(t) e
im⌦
'
t
±`m
(t, r) =
X
n
R
±`mn
(r) e
�i!t
! = m⌦
'
+ n⌦
r
a, b, . . . 2 {t, r}
A,B, . . . 2 {✓,'}
(t, r, ✓,') ! ⇥t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)
⇤
±`m
(t, r) =
X
n
R
±`mn
(r) e
�i!t
! = m⌦
'
+ n⌦
r
a, b, . . . 2 {t, r}
A,B, . . . 2 {✓,'}
(t, r, ✓,') ! ⇥t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)
⇤
Radially periodic
S
`m
(t, r) = G
`m
(t) �[r � r
p
(t)] + F
`m
(t) �
0[r � r
p
(t)]
¯
±22
(�)
±22
(�)
C
±`mn
=
1
W
`mn
T
r
ZT
r
0
"1
f
p
ˆ
R
⌥`mn
(r
p
)
¯
G
`m
(t)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(r
p
)� 1
f
p
d
ˆ
R
⌥`mn
(r
p
)
dr
!¯
F
`m
(t)
#e
in⌦r
t
dt
¯
G
`m
(t) ⌘ G
`m
(t) e
im⌦
'
t
,
¯
F
`m
(t) ⌘ F
`m
(t) e
im⌦
'
t
Master equation source term
Homogeneous solutions
C
±`mn
=
1
W
`mn
T
r
ZT
r
0
"1
f
p
ˆ
R
⌥`mn
(r
p
)
¯
G
`m
(t)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(r
p
)� 1
f
p
d
ˆ
R
⌥`mn
(r
p
)
dr
!¯
F
`m
(t)
#e
in⌦r
t
dt
¯
G
`m
(t) ⌘ G
`m
(t) e
im⌦
'
t
,
¯
F
`m
(t) ⌘ F
`m
(t) e
im⌦
'
t
±`m
(t, r) =
X
n
R
±`mn
(r) e
�i!t
! = m⌦
'
+ n⌦
r
a, b, . . . 2 {t, r}
A,B, . . . 2 {✓,'}
Manifestly radially periodic
Removing the azimuthal frequency dependence from
the master function makes it periodic
±`m
(t, r) =
X
n
R
±`mn
(r) e
�i!t
! = m⌦
'
+ n⌦
r
a, b, . . . 2 {t, r}
A,B, . . . 2 {✓,'}
(t, r, ✓,') ! ⇥t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)
⇤
-�
-�
�
�
�π�
π�π�
�π
¯
±22
(�)
±22
(�)
C
±`mn
=
1
W
`mn
T
r
ZT
r
0
"1
f
p
ˆ
R
⌥`mn
(r
p
)
¯
G
`m
(t)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(r
p
)� 1
f
p
d
ˆ
R
⌥`mn
(r
p
)
dr
!¯
F
`m
(t)
#e
in⌦r
t
dt
¯
G
`m
(t) ⌘ G
`m
(t) e
im⌦
'
t
,
¯
F
`m
(t) ⌘ F
`m
(t) e
im⌦
'
t
±`m
(t, r) =
X
n
R
±`mn
(r) e
�i!t
p = 10
e = 0.5
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
¯
±22
(�)
±22
(�)
C
±`mn
=
1
W
`mn
T
r
ZT
r
0
"1
f
p
ˆ
R
⌥`mn
(r
p
)
¯
G
`m
(t)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(r
p
)� 1
f
p
d
ˆ
R
⌥`mn
(r
p
)
dr
!¯
F
`m
(t)
#e
in⌦r
t
dt
¯
G
`m
(t) ⌘ G
`m
(t) e
im⌦
'
t
,
¯
F
`m
(t) ⌘ F
`m
(t) e
im⌦
'
t
±`m
(t, r) =
X
n
R
±`mn
(r) e
�i!t
-�
�
�
�π�
π�π�
�π
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
h
21r
(�)
¯
h
21r
(�)
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
h
21r
(�)
¯
h
21r
(�)
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
“Barred” versions of the metric perturbation
amplitudes pick up counter terms
h
`m,±t
=
f
2
@
r
�r
±`m
�, h
`m,±r
=
r
2f
�@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�r
¯
±`m
�,
¯
h
`m,±r
=
r
2f
�@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦
'
t
¯
±`m
(t, r) ⌘
±`m
(t, r)e
im⌦
'
t
=
X
n
R
±`mn
(r) e
�in⌦
r
t
X
1
! M
r
p
X
2
! (!r
p
)
2
π�
π�π�
�π
-���
-���
-���
���
���
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
h
21r
(�)
¯
h
21r
(�)
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
π�
π�π�
�π
-���
-���
���
���
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
h
21r
(�)
¯
h
21r
(�)
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
p = 10
e = 0.5
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
Quantities evaluated on the worldline are strictly
periodic in the radial motion
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(t
p
, r
p
, ✓
p
,'
p
) =
X
`,m
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦
'
t
Y
`m
(✓
p
, 0)e
im⌦
'
t
p
tt
(t
p
, r
p
, ✓
p
,'
p
) =
X
`,m
¯
h
`m
tt
(t
p
, r
p
)
¯
Y
`m
¯
Y
`m
(✓
p
, 0)Y
`m
(✓
p
, 0)
h
`m,±t
=
f
2
@
r
�r
±`m
�, h
`m,±r
=
r
2f
�@
t
±`m
�
Dependent on four variables
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
h
21r
(�)
¯
h
21r
(�)
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
h
21r
(�)
¯
h
21r
(�)
Evaluated at the particle
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(t) =
X
`,m
⇣¯
h
`m
tt
(t, r
p
)e
�im⌦
'
t
⌘⇣Y
`m
[✓
p
,�'(t)]e
im⌦
'
t
⌘
p
tt
(t) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦
'
t
h
`m,±t
=
f
2
@
r
�r
±`m
�, h
`m,±r
=
r
2f
�@
t
±`m
�
(t, r, ✓,') ! ⇥t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)
⇤
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h¯
h
`m
tt
(t, r
p
)e
�im⌦
'
t
i·hY
`m
(✓
p
, 0)e
im�'(t)
e
im⌦
'
t
i
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦
'
t
�π�
π�π�
�π
�����
�����
�����
�����Re
hp
22
tt
(�)
i
Im
hp
22
tt
(�)
i
t
p
(�) =
Z�
0
p
2
M
(p� 2� 2e cos�
0)(1 + e cos�
0)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
0
#1/2
d�
0
p = 10
e = 0.5
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
Outline
Orbits
Homogeneous solutions
Particular solutions
Black Hole Perturbation Theory
Formalism
Previous work
Analytic PN solutions
Previous work
Numeric PN solutions
There has been a lot of previous work, but here are a
couple highlights
Mano, Suzuki & Takasugi
Ganz, Hikida, Nakano, Sago &
Tanaka
Sago & Fujita
1996 1997 2007 2015
Mano & Takasugi
General orbits on Kerr
4PN & 𝑒6
Outline
Orbits
Homogeneous solutions
Particular solutions
Black Hole Perturbation Theory
Formalism
Previous work
Analytic PN solutions
Previous work
Numeric PN solutions
The fundamental frequencies can be expanded in the
gauge invariant PN parameter 𝑥
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
¯
±`m
(t, r) ⌘ ±`m
(t, r)e
im⌦'t
=
X
n
R
±`mn
(r) e
�in⌦rt
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
¯
h
`m,±t
=
f
2
@
r
�
r
¯
±`m
�
¯
h
`m,±r
=
r
2f
�
@
t
¯
±`m
� im⌦
'
¯
±`m
�
¯
±`m
(t, r) =
±`m
(t, r)e
im⌦'t
Integrate order-by-order
⌦'
=1
M
1� e
2
p
!3/2 "1 + 3
e
2
p
+O⇣p
�2⌘#
x =�M⌦
'
�2/3
T
r
'(t)
'(Tr
)
�'(t)
X1 =M
r
X2 = (!r)2
1
⌦' =1
M
1� e2
p
!3/2 "1 + 3
e2
p+O
⇣p�2⌘#
Tr
'(t)
'(Tr)
�'(t)
X1 =M
rX2 = (!r)2
X1 ⇠ X2 ⌧ 1
1
𝑥 as an expansion in 1/𝑝
p =
1� e
2
x
+ 2e
2
+O⇣x
1
⌘
⌦
r
=
x
3/2
M
1� 3
1� e
2
x +O(x
2
)
�
⌦
'
=
x
3/2
M
dt
p
d�
=
p
2
M
(p� 2� 2e cos�)(1 + e cos�)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
#1/2
1
Invert the expansion
⌦
r
=
x
3/2
M
1� 3
1� e
2
x +O(x
2
)
�
⌦
'
=
x
3/2
M
dt
p
d�
=
p
2
M
(p� 2� 2e cos�)(1 + e cos�)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
#1/2
'
p
(�) =
✓4p
p� 6� 2e
◆1/2
F
✓�
2
�����4e
p� 6� 2e
◆1
⌦
r
=
x
3/2
M
1� 3
1� e
2
x +O(x
2
)
�
⌦
'
=
x
3/2
M
dt
p
d�
=
p
2
M
(p� 2� 2e cos�)(1 + e cos�)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
#1/2
'
p
(�) =
✓4p
p� 6� 2e
◆1/2
F
✓�
2
�����4e
p� 6� 2e
◆
1
Fundamental frequencies
Homogeneous solutions are found by expanding in
two separate limits
Tr
'(t)
'(Tr)
�'(t)
X1 =M
rX2 = (!r)2
X1 ⇠ X2 ⌧ 1
t
0 t Tr
rmin ! rmin1
Expansion variables Example, 𝑙 = 2, through 1PN
Tr
'(t)
'(Tr)
�'(t)
X1 =M
rX2 = (!r)2
X1 ⇠ X2 ⌧ 1
R+ = � i
16X2c2 � i(10X1 +X2)
96X2c0 +O
⇣c�2
⌘
1R� =
i
384X41pX2
c9 � ipX2
5376X41
c7 +O⇣c5⌘
t
0 t Tr
rmin ! rmin
` = 2
⌦r =2⇡
Tr
⌦' ='(Tr)
Tr
�'(t) = '(t)� ⌦'t
2
"d2
dr2⇤+ !2 � V`(r)
#R`!(r) = 0
fd
dr
✓fd
dr
◆+ !2 � Vl(r)
�Xlm!(r) = 0
3
Odd parity, homogenous
equation
Use expressions for the position and the
frequency, expanded in both 𝑥 and 𝑒
'
p
(�)
C
±=
1
WT
r
Z 2⇡
0
"1
f
p
X
⌥¯
G(t) +
✓2M
r
2p
f
2p
X
⌥ � 1
f
p
dX
⌥
dr
◆¯
F (t)
#e
in⌦r
t
dt
p
d�
d�
�'(�)
r
p
(�) =
hM � eM cos(�) +O �
e
2� i
x
�1
+
h2e
2M � 2e
3M cos(�) +O �
e
4� i
x
0+O �
x
1�.
(p, e) = (10, 0.5)
! =
(m + n)
M
x
3/2
+
"�3n
M
� 3ne
2
M
+O⇣e
3
⌘#x
5/2
+O⇣x
3
⌘.
R
+
= � i
16!
2
r
2
p
c
2 � 1
96
i
10M
!
2
r
3
p
+ 1
!c
0
+O⇣c
�2
⌘
R
�=
ir
3
p
384M
4
!
c
9 � i!r
5
p
5376M
4
c
7
+O⇣c
5
⌘
R
+=
� i
16(m + n)
2� i cos(�)e
8(m + n)
2+O �e2�
�x
�1
+
"�i
�m
3+ 3m
2n + 3mn
2+ 10m + n
3+ 46n
�
96(m + n)
3� ie(5m + 17n) cos(�)
16(m + n)
3+O �e2�
#x
0+O �x1�
Spectrum from source’s Fourier seriesR
+
= � i
16!
2
r
2
p
c
2 � 1
96
i
10M
!
2
r
3
p
+ 1
!c
0
+O⇣c
�2
⌘
R
�=
ir
3
p
384M
4
!
c
9 � i!r
5
p
5376M
4
c
7
+O⇣c
5
⌘
R
+=
� i
16(m + n)
2� i cos(�)e
8(m + n)
2+O �e2�
�x
�1
+
"�i
�m
3+ 3m
2n + 3mn
2+ 10m + n
3+ 46n
�
96(m + n)
3� ie(5m + 17n) cos(�)
16(m + n)
3+O �e2�
#x
0+O �x1�
R
±`mn
(r) ⌘ C
±`mn
ˆ
R
±`mn
(r)
±`m
(t, r) =
X
n
C
±`mn
ˆ
R
±`mn
(r) e
�i!t
r > 2M
R
�=
i
384(m+ n)
� i cos(�)e
128(m+ n)
+O �e
2��
x
�9/2
+
"�i
(m+ n)
3 � 42n
5376(m+ n)
2+
i
�5m
3+ 15nm
2+ 15n
2m+ 5n
3 � 126n
�cos(�)e
5376(m+ n)
2+O �
e
2�#x
�7/2+O
⇣x
�5/2⌘
¯
G
o
2m =
p5⇡µ
3M
(m� 3)
s(2�m)!
(2 +m)!
P
m
3
(h2 +
�e
i�
(2� 2m) + 2e
�i�
(m+ 1)
�e+O �
e
2� i
x
3/2
+
h� 1 +
�2e
�i�
(m� 2)� 2e
i�
(m+ 2)
�e+O �
e
2� i
x
5/2+O
⇣x
7/2⌘)
R
+=
� i
16(m+ n)
2� i cos(�)e
8(m+ n)
2+O �
e
2��
x
�1
+
"�i
�m
3+ 3m
2n+ 3mn
2+ 10m+ n
3+ 46n
�
96(m+ n)
3� ie(5m+ 17n) cos(�)
16(m+ n)
3+O �
e
2�#x
0+O �
x
1�
R
±`mn
(r) ⌘ C
±`mn
ˆ
R
±`mn
(r)
±`m
(t, r) =
X
n
C
±`mn
ˆ
R
±`mn
(r) e
�i!t
r > 2M
Homogeneous solutions, odd parity, 𝑙=2, generic 𝑚,𝑛
Normalized frequency domain solutions are found for
generic 𝑚 and 𝑛R
+2mn = µ
s(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(�8 ((m� 3)P
m3 )
3n
+
✓8(m� 3)(2m� n)
3 (n
2 � 1)
� 16(m� 3) cos(�)
3n
◆P
m3 e+O �e2�
�px
+
"� 2 ((m� 3) (5m
2+ 10nm+ 5n
2+ 98)P
m3 )
63n
+
2(m� 3) (10m
3+ 33nm
2+ 36n
2m+ 511m+ 13n
3 � 287n)
63 (n
2 � 1)
+
4(m� 3) (3m
2+ 6nm+ 3n
2 � 56) cos(�)
21n
!P
m3 e+O �e2�
#x
3/2+O �x2
�)
C
+2mn = µ
s(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(�128i(m� 3)(m+ n)
2P
m3
3n
+
128i(m� 3)(2m� n)(m+ n)
2P
m3 e
3 (n
2 � 1)
+O �e2��x
3/2
+
"32i(m� 3) (3m
4+ 12nm
3+ 2 (9n
2+ 7)m
2+ 4n (3n
2+ 49)m+ n
2(3n
2+ 182))P
m3
21n
� 32i(m� 3)(m+ n) (6m
4+ 9nm
3 � (9n
2+ 77)m
2 � 7n (3n
2 � 44)m� n
2(9n
2+ 119))P
m3 e
21 (n
2 � 1)
+O �e2�#x
5/2+O �x3
�)
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
R
+2mn = µ
s(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(�8 ((m� 3)P
m3 )
3n
+
✓8(m� 3)(2m� n)
3 (n
2 � 1)
� 16(m� 3) cos(�)
3n
◆P
m3 e+O �e2�
�px
+
"� 2 ((m� 3) (5m
2+ 10nm+ 5n
2+ 98)P
m3 )
63n
+
2(m� 3) (10m
3+ 33nm
2+ 36n
2m+ 511m+ 13n
3 � 287n)
63 (n
2 � 1)
+
4(m� 3) (3m
2+ 6nm+ 3n
2 � 56) cos(�)
21n
!P
m3 e+O �e2�
#x
3/2+O �x2
�)
C
+2mn = µ
s(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(�128i(m� 3)(m+ n)
2P
m3
3n
+
128i(m� 3)(2m� n)(m+ n)
2P
m3 e
3 (n
2 � 1)
+O �e2��x
3/2
+
"32i(m� 3) (3m
4+ 12nm
3+ 2 (9n
2+ 7)m
2+ 4n (3n
2+ 49)m+ n
2(3n
2+ 182))P
m3
21n
� 32i(m� 3)(m+ n) (6m
4+ 9nm
3 � (9n
2+ 77)m
2 � 7n (3n
2 � 44)m� n
2(9n
2+ 119))P
m3 e
21 (n
2 � 1)
+O �e2�#x
5/2+O �x3
�)
FD extended homogeneous solutions, odd parity, 𝑙=2, generic 𝑚,𝑛
C
±`mn
=
1
W
`mn
T
r
Z 2⇡
0
"
1
f
p
ˆ
R
⌥`mn
(�)
¯
G
`m
(�)+
2M
r
2p
f
2p
ˆ
R
⌥`mn
(�)� 1
f
p
d
ˆ
R
⌥`mn
(�)
dr
!
¯
F
`m
(�)
#
e
in⌦rtp(�) dtp
d�
d�
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �e2�i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �e2�i
x
5/2+O �x7/2
�
)
x
3/2
The finite-order 𝒆 expansion truncates the sum over
harmonics
¯
±`m(t, r) =
X
n
R
±`mn(r) e
�in⌦rt
n R
+2mn(�)
�3 0
�2 �µ3
q
⇡(2�m)!5(2+m)! (m� 3) (4m
2 �m� 2)P
m3 e
2
�1 �4µ3
q
⇡(2�m)!5(2+m)! (m� 3)(�2m� 1)P
m3
h
e+ 2e
2cos(�)
i
0 �8µ3
q
⇡(2�m)!5(2+m)! (m� 3)P
m3
n
1 + 2e cos(�) + e
2⇥
2�m
2+ cos
2(�)
⇤
o
1 �4µ3
q
⇡(2�m)!5(2+m)! (m� 3)(2m� 1)P
m3
h
e+ 2e
2cos(�)
i
2 �µ3
q
⇡(2�m)!5(2+m)! (m� 3) (4m
2+m� 2)P
m3 e
2
3 0
R
+2mn = µ
s
(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(
�8 ((m� 3)P
m3 )
3n
+
✓
8(m� 3)(2m� n)
3 (n
2 � 1)
� 16(m� 3) cos(�)
3n
◆
P
m3 e+O �e2�
�px
+
"
� 2 ((m� 3) (5m
2+ 10nm+ 5n
2+ 98)P
m3 )
63n
+
2(m� 3) (10m
3+ 33nm
2+ 36n
2m+ 511m+ 13n
3 � 287n)
63 (n
2 � 1)
+
4(m� 3) (3m
2+ 6nm+ 3n
2 � 56) cos(�)
21n
!
P
m3 e+O �e2�
#
x
3/2+O �x2
�
)
C
+2mn = µ
s
(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(
�128i(m� 3)(m+ n)
2P
m3
3n
+
128i(m� 3)(2m� n)(m+ n)
2P
m3 e
3 (n
2 � 1)
+O �e2��
x
3/2
+
"
32i(m� 3) (3m
4+ 12nm
3+ 2 (9n
2+ 7)m
2+ 4n (3n
2+ 49)m+ n
2(3n
2+ 182))P
m3
21n
� 32i(m� 3)(m+ n) (6m
4+ 9nm
3 � (9n
2+ 77)m
2 � 7n (3n
2 � 44)m� n
2(9n
2+ 119))P
m3 e
21 (n
2 � 1)
+O �e2�#
x
5/2+O �x3
�
)
p
`m,±µ⌫
(�)
@
t
p
`m,±µ⌫
(�)
@
r
p
`m,±µ⌫
(�)
¯
±`m
(�) =
X
n
R
±`mn
(�) e
�in⌦rtp(�)
n R
+2mn
(�)
�3 0
�2 �µ
3
q
⇡(2�m)!5(2+m)! (m� 3) (4m
2 �m� 2)P
m
3 e
2
�1 �4µ3
q
⇡(2�m)!5(2+m)! (m� 3)(�2m� 1)P
m
3
h
e+ 2e
2cos(�)
i
0 �8µ3
q
⇡(2�m)!5(2+m)! (m� 3)P
m
3
n
1 + 2e cos(�) + e
2⇥
2�m
2+ cos
2(�)
⇤
o
1 �4µ3
q
⇡(2�m)!5(2+m)! (m� 3)(2m� 1)P
m
3
h
e+ 2e
2cos(�)
i
2 �µ
3
q
⇡(2�m)!5(2+m)! (m� 3) (4m
2+m� 2)P
m
3 e
2
3 0
R
+2mn
= µ
s
(2�m)!
(2 +m)!
sin(n⇡)p5⇡
(
�8 ((m� 3)P
m
3 )
3n
+
✓
8(m� 3)(2m� n)
3 (n
2 � 1)
� 16(m� 3) cos(�)
3n
◆
P
m
3 e+O �e2��p
x
+
"
� 2 ((m� 3) (5m
2+ 10nm+ 5n
2+ 98)P
m
3 )
63n
+
2(m� 3) (10m
3+ 33nm
2+ 36n
2m+ 511m+ 13n
3 � 287n)
63 (n
2 � 1)
+
4(m� 3) (3m
2+ 6nm+ 3n
2 � 56) cos(�)
21n
!
P
m
3 e+O �e2�#
x
3/2+O �x2
�
)
The retarded metric perturbation is computed only once
for each 𝑙 mode
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
`m
(�),
¯
F
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
±`m
(�)
@
t
¯
±`m
(�), @
2t
¯
±`m
(�)
@
r
¯
±`m
(�), @
2r
¯
±`m
(�)
.
.
.
p
`m
µ⌫
(�)
@
t
p
`m
µ⌫
(�)
@
r
p
`m
µ⌫
(�)
Orbit quantities
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
`m
(�),
¯
F
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
±`m
(�)
@
t
¯
±`m
(�), @
2t
¯
±`m
(�)
@
r
¯
±`m
(�), @
2r
¯
±`m
(�)
.
.
.
p
`m
µ⌫
(�)
@
t
p
`m
µ⌫
(�)
@
r
p
`m
µ⌫
(�)
Norm. coeffs.
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
`m
(�),
¯
F
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
±`m
(�)
@
t
¯
±`m
(�), @
2t
¯
±`m
(�)
@
r
¯
±`m
(�), @
2r
¯
±`m
(�)
.
.
.
p
`m
µ⌫
(�)
@
t
p
`m
µ⌫
(�)
@
r
p
`m
µ⌫
(�)
FD particular sols. Metric perturbation
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
p
`m,±µ⌫
(�)
@
t
p
`m,±µ⌫
(�)
@
r
p
`m,±µ⌫
(�)
TD particular sols.
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
p
`m,±µ⌫
(�)
@
t
p
`m,±µ⌫
(�)
@
r
p
`m,±µ⌫
(�)
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
`m
(�),
¯
F
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
±`m
(�)
@
t
¯
±`m
(�), @
2t
¯
±`m
(�)
@
r
¯
±`m
(�), @
2r
¯
±`m
(�)
.
.
.
p
`m
µ⌫
(�)
@
t
p
`m
µ⌫
(�)
@
r
p
`m
µ⌫
(�)
Homog. sols.
Source terms
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
ˆ
R
e/o,±`mn
(�)
@
r
ˆ
R
e/o,±`mn
(�)
@
2r
ˆ
R
e/o,±`mn
(�)
¯
G
e/o
`m
(�),
¯
F
e/o
`m
(�)
C
e/o,±`mn
R
e/o,±`mn
(�)
@
r
R
e/o,±`mn
(�)
@
2r
R
e/o,±`mn
(�)
¯
e/o,±`m
(�)
@
t
¯
e/o,±`m
(�), @
2t
¯
e/o,±`m
(�)
@
r
¯
e/o,±`m
(�), @
2r
¯
e/o,±`m
(�)
.
.
.
p
`m,±µ⌫
(�)
@
t
p
`m,±µ⌫
(�)
@
r
p
`m,±µ⌫
(�)
Metric perturbation expressions are remarkably simple
p
22,+tt
=
µ
M
3
4
+
✓3e
�i�
8
+
3e
i�
8
◆e +O �
e
2��
x
+
µ
M
�149
56
+
✓�229
112
e
�i� � 437e
i�
112
◆e +O �
e
2��
x
2+O �
x
3�
p
22,�tt
=
µ
M
3
4
+
✓3e
�i�
8
+
3e
i�
8
◆e +O �
e
2��
x
+
µ
M
�149
56
+
✓�509
112
e
�i� � 157e
i�
112
◆e +O �
e
2��
x
2+O �
x
3�
�π�
π�π�
�π
�����
�����
�����
�����
Re
hp
22
tt
(�)
i
Im
hp
22
tt
(�)
i
t
p
(�) =
Z�
0
p
2
M
(p� 2� 2e cos�
0)(1 + e cos�
0)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
0
#1/2
d�
0
r
p
(�) =
hM �M cos(�)e +O
⇣e
2
⌘ ix
�1
+
h2e
2
M � 2e
3
M cos(�) +O⇣e
4
⌘ ix
0
+O⇣x
1
⌘.
(p, e) = (10, 0.5)
p = 10
e = 0.5
x = 0.08
p =
1� e
2
x
+ 2e
2
+O⇣x
1
⌘
1
We see convergence in 𝑥 for small values of 𝑒
π�
π �π�
�π
-��
-��
-�
-�
-�
-��� �����
�����
p = 1000
e = 0.01
x ⇡ 0.001
log
10
hp
22
tt
res.
i
p
22
tt
(�)
Re
hp
22
tt
(�)
i
Im
hp
22
tt
(�)
i
t
p
(�) =
Z�
0
p
2
M
(p� 2� 2e cos�
0)(1 + e cos�
0)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
0
#1/2
d�
0
p = 1000
e = 0.01
x ⇡ 0.001
log
10
hp
22
tt
residuals
i
p
22
tt
(�)
Re
hp
22
tt
(�)
i
Im
hp
22
tt
(�)
i
t
p
(�) =
Z�
0
p
2
M
(p� 2� 2e cos�
0)(1 + e cos�
0)
2
"(p� 2)
2 � 4e
2
p� 6� 2e cos�
0
#1/2
d�
0
Sometimes the convergence with 𝑒 stalls
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
residuals
⇤
log10
⇥
p
22,�tr
residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
⌦
'
,⌦
r
E ,Jr
p
(�), r
p
(�), t
p
(�), �'(�)
������������ ������ � � � � � � � � � ��
π�
π �π�
�π
-�
-�
-�
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
π�
π�π�
�π
-�
-�
-�
-�
-�
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
e
9
e
5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
For small values of 𝑒 the convergence continues
π�
π�π�
�π
-��
-��
-�
������������ ������ � � � � � � � � � ��
π�
π �π�
�π
-��
-��
-��
-�
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
residuals
⇤
log10
⇥
p
22,�tr
residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
p = 10
6
e = 0.01
x = 10
�6
p = 10
6
e = 0.2
x = 0.96⇥ 10
�6
log10
⇥
p
22,+tr
(�) residuals
⇤
log10
⇥
p
22,�tr
(�) residuals
⇤
¯
G
`m
(�) = 8
p⇡
µ
M
(`�m+ 1)
(`� 1)`(`+ 1)(`+ 2)
s
(2`+ 1)(`�m)!
(m+ `)!
P
m
`+1
⇥(
h
� 2 +
�
2e
i�
(m� 1)� 2e
�i�
(m+ 1)
�
e+O �
e
2�
i
x
3/2
+
h
1 +
�
e
�i�
(4� 2m) + 2e
i�
(m+ 2)
�
e+O �
e
2�
i
x
5/2+O �
x
7/2�
)
x
3/2
R
±`mn
(�) = C
±`mn
ˆ
R
±`mn
(�)
p = 10
e = 0.5
Outline
Orbits
Homogeneous solutions
Particular solutions
Black Hole Perturbation Theory
Formalism
Previous work
Analytic PN solutions
Previous work
Numeric PN solutions
Here are specific numerical highlights
Mano, Suzuki & Takasugi
1996 1997
Mano & Takasugi
2014
Shah
Kerr fluxes
Shah, Friedman, & Whiting
Redshift invariant
Outline
Orbits
Homogeneous solutions
Particular solutions
Black Hole Perturbation Theory
Formalism
Previous work
Analytic PN solutions
Previous work
Numeric PN solutions
Our new code can solve the first order field equations
to hundreds of digits
Mathematica for arbitrary precision
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
⌧
dE
dt
�
N
L7/2x7/2
+ x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
p = 1000
p = 10
20
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
⌧
dE
dt
�
N
L7/2x7/2
+ x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
p = 1000
p = 10
20
e = 0.1
(1)
"d
2
dr
2⇤+ !
2 � V
`
(r)
#R
`mn
(r) = Z
`mn
(r)
! ⌘ !
mn
= m⌦
'
+ n⌦
r
`m
(t, r) =
1X
n=�1R
`mn
(r) e
�i!t
S
`m
(t, r) =
1X
n=�1Z
`mn
(r) e
�i!t
Energy flux from an eccentric binary is known to 3PN
8
Once the C
±lmn
are determined, the energy fluxes at infinity can be calculated using
⌧
dE
dt
�
=X
lmn
!
2
64⇡
(l + 2)!
(l � 2)!|C+
lmn
|2, (3.26)
given our initial unit normalization of the modes X±lmn
.
IV. CONFIRMING ECCENTRIC ORBIT FLUXES THROUGH 3PN RELATIVE ORDER
The average energy and angular momentum fluxes from an eccentric binary are known to 3PN relative order[49–51] (see also the review by Blanchet [20]). The expressions are given in terms of three parameters; e.g., the
gauge-invariant post-Newtonian compactness parameter x ⌘ (M⌦'
)2/3, the eccentricity, and the symmetric massratio ⌫ = m
1
m
2
/(m1
+m
2
)2 ' µ/M . In this paper we ignore contributions that are higher order in the mass ratiothan O(⌫2), as these would require a second-order GSF calculation to reach. The energy flux is composed of a set ofeccentricity-dependent coe�cients. Its form through 3PN relative order is
⌧
dE
dt
�
3PN
=32
5
⇣
µ
M
⌘
2
x
5
⇣
I0
+ x I1
+ x
3/2 K3/2
+ x
2 I2
+ x
5/2 K5/2
+ x
3 I3
+ x
3 K3
⌘
. (4.1)
The In
are instantaneous flux functions [of eccentricity and (potentially) log(x)] that have known closed-form ex-pressions (summarized below). The K
n
coe�cients are hereditary, or tail, contributions (without apparently closedforms). In a subsequent section, we derive new expansions for these hereditary terms, which are useful at minimumfor BHP/PN comparisons.
A. Known instantaneous energy flux terms
For later reference and use, we list here the instantaneous energy flux functions, expressed in modified harmonic(MH) gauge [20, 49, 51] and in terms of e
t
, a particular definition of eccentricity (“time eccentricity”) used in thequasi-Keplerian (QK) representation [52] of the orbit (see also [20, 49–51, 53–55])
I0
=1
(1� e
2
t
)7/2
✓
1 +73
24e
2
t
+37
96e
4
t
◆
, (4.2)
I1
=1
(1� e
2
t
)9/2
✓
�1247
336+
10475
672e
2
t
+10043
384e
4
t
+2179
1792e
6
t
◆
, (4.3)
I2
=1
(1� e
2
t
)11/2
✓
�203471
9072� 3807197
18144e
2
t
� 268447
24192e
4
t
+1307105
16128e
6
t
+86567
64512e
8
t
◆
+1
(1� e
2
t
)5
✓
35
2+
6425
48e
2
t
+5065
64e
4
t
+185
96e
6
t
◆
,
(4.4)
I3
=1
(1� e
2
t
)13/2
✓
2193295679
9979200+
20506331429
19958400e
2
t
� 3611354071
13305600e
4
t
+4786812253
26611200e
6
t
+21505140101
141926400e
8
t
� 8977637
11354112e
10
t
◆
+1
(1� e
2
t
)6
✓
�14047483
151200+
36863231
100800e
2
t
+759524951
403200e
4
t
+1399661203
2419200e
6
t
+185
48e
8
t
◆
+1712
105log
"
x
x
0
1 +p
1� e
2
t
2(1� e
2
t
)
#
F (et
),
(4.5)
where the function F (et
) in Eqn. (4.5) has the following closed-form [49]
F (et
) =1
(1� e
2
t
)13/2
✓
1 +85
6e
2
t
+5171
192e
4
t
+1751
192e
6
t
+297
1024e
8
t
◆
. (4.6)
Peters & Mathews, 1963
8
Once the C
±lmn
are determined, the energy fluxes at infinity can be calculated using
⌧
dE
dt
�
=X
lmn
!
2
64⇡
(l + 2)!
(l � 2)!|C+
lmn
|2, (3.26)
given our initial unit normalization of the modes X±lmn
.
IV. CONFIRMING ECCENTRIC ORBIT FLUXES THROUGH 3PN RELATIVE ORDER
The average energy and angular momentum fluxes from an eccentric binary are known to 3PN relative order[49–51] (see also the review by Blanchet [20]). The expressions are given in terms of three parameters; e.g., the
gauge-invariant post-Newtonian compactness parameter x ⌘ (M⌦'
)2/3, the eccentricity, and the symmetric massratio ⌫ = m
1
m
2
/(m1
+m
2
)2 ' µ/M . In this paper we ignore contributions that are higher order in the mass ratiothan O(⌫2), as these would require a second-order GSF calculation to reach. The energy flux is composed of a set ofeccentricity-dependent coe�cients. Its form through 3PN relative order is
⌧
dE
dt
�
3PN
=32
5
⇣
µ
M
⌘
2
x
5
⇣
I0
+ x I1
+ x
3/2 K3/2
+ x
2 I2
+ x
5/2 K5/2
+ x
3 I3
+ x
3 K3
⌘
. (4.1)
The In
are instantaneous flux functions [of eccentricity and (potentially) log(x)] that have known closed-form ex-pressions (summarized below). The K
n
coe�cients are hereditary, or tail, contributions (without apparently closedforms). In a subsequent section, we derive new expansions for these hereditary terms, which are useful at minimumfor BHP/PN comparisons.
A. Known instantaneous energy flux terms
For later reference and use, we list here the instantaneous energy flux functions, expressed in modified harmonic(MH) gauge [20, 49, 51] and in terms of e
t
, a particular definition of eccentricity (“time eccentricity”) used in thequasi-Keplerian (QK) representation [52] of the orbit (see also [20, 49–51, 53–55])
I0
=1
(1� e
2
t
)7/2
✓
1 +73
24e
2
t
+37
96e
4
t
◆
, (4.2)
I1
=1
(1� e
2
t
)9/2
✓
�1247
336+
10475
672e
2
t
+10043
384e
4
t
+2179
1792e
6
t
◆
, (4.3)
I2
=1
(1� e
2
t
)11/2
✓
�203471
9072� 3807197
18144e
2
t
� 268447
24192e
4
t
+1307105
16128e
6
t
+86567
64512e
8
t
◆
+1
(1� e
2
t
)5
✓
35
2+
6425
48e
2
t
+5065
64e
4
t
+185
96e
6
t
◆
,
(4.4)
I3
=1
(1� e
2
t
)13/2
✓
2193295679
9979200+
20506331429
19958400e
2
t
� 3611354071
13305600e
4
t
+4786812253
26611200e
6
t
+21505140101
141926400e
8
t
� 8977637
11354112e
10
t
◆
+1
(1� e
2
t
)6
✓
�14047483
151200+
36863231
100800e
2
t
+759524951
403200e
4
t
+1399661203
2419200e
6
t
+185
48e
8
t
◆
+1712
105log
"
x
x
0
1 +p
1� e
2
t
2(1� e
2
t
)
#
F (et
),
(4.5)
where the function F (et
) in Eqn. (4.5) has the following closed-form [49]
F (et
) =1
(1� e
2
t
)13/2
✓
1 +85
6e
2
t
+5171
192e
4
t
+1751
192e
6
t
+297
1024e
8
t
◆
. (4.6)
“Enhances” flux from Hulse-Taylor pulsar (𝑒=0.62) by factor of 12
We confirm previously known PN parameters through
3PN
�80
�70
�60
�50
�40
�30
�20
�10
0
10
0 0.02 0.04 0.06 0.08 0.1
log
(P
Nresiduals)
eccentricity e
Peters-Mathews Flux
hEi � Peters-Mathews = O(1PN)
hEi � 1PN = O(1.5PN)
hEi � 1.5PN = O(2PN)
hEi � 2PN = O(2.5PN)
hEi � 2.5PN = O(3PN)
hEi � 3PN = O(3.5PN)
This was hard
p = 10
20
p
µ⌫
(�),
@p
µ⌫
@t
(�),
@p
µ⌫
@r
(�)
0 ` 10
4 PN, e
10
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
We found new parameters through 7PN
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
⌧
dE
dt
�
N
L7/2x7/2
+ x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
p = 1000
p = 10
20
p
µ⌫
(�),
@p
µ⌫
@t
(�),
@p
µ⌫
@r
(�)
0 ` 10
4 PN, e
10
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2"
1 +
3
�
1� e
2�
p
+O �
p
�2�
#
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O⇣
x
1/2⌘
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t
⇤ ·h
Y
`m
(✓
p
, 0)e
im�'(t)e
im⌦'tp
i
Circular orbit limit Guessing the correct singular behavior extends range of validity
We spanned the two dimensional space of orbits
𝑒 = 0 𝑒 = 0.1
𝑝 = 1010
𝑝 = 1035
1683 orbits
33 eccentricities
51 radii
Each orbit involves solving for hundreds of modes
�300
�250
�200
�150
�100
-100 -50 0 50 100
log
D E1 `m
n
E
Harmonic number n
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
⌧
dE
dt
�
N
L7/2x7/2
+ x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
p = 1000
p = 10
20
e = 0.1
(1)
PN parameters involve sums of trancendentals
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
⌧
dE
dt
�
N
L7/2x7/2
+ x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
p = 1000
p = 10
20
p
µ⌫
(�),
@p
µ⌫
@t
(�),
@p
µ⌫
@r
(�)
0 ` 10
4 PN, e
10
Thanks to Nathan Johnson-McDaniel
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
⌧
dE
dt
�
N
L7/2x7/2
+ x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
p = 1000
p = 10
20
p
µ⌫
(�),
@p
µ⌫
@t
(�),
@p
µ⌫
@r
(�)
0 ` 10
4 PN, e
10
Found with PSLQ algorithm
New PN parameters are valid well outside the regime
where they were found
Higher order PN residuals
0.60
0.90
�1.3
�1.6
�2.5
�2.2
�4
�4.3
�4.2
�4.5
�5.5
�5.2
�7.1
�7.4
�7.2
�7.5
�8.5
�8.2
�10.6
�10.9
�10.2
�10.5
0 0.2 0.4 0.6
Eccentricity e
0 0.2 0.4 0.6
�11.5
�11.2
Eccentricity e
I0 + O(x)
hEi/hEiC
xI1 + O(x3/2)
0
BBBBBBBBBBBBB@
hEi � I0
1
CCCCCCCCCCCCCA
/hEiC
x
3/2K3/2 + O(x2)
0
BBBBBBBBBBBBB@
hEi � hEi1PN
1
CCCCCCCCCCCCCA
/hEiC
x
2I2 + O(x5/2)
0
BBBBBBBBBBBBB@
hEi � hEi1.5PN
1
CCCCCCCCCCCCCA
/hEiC
x
5/2I5/2 + O(x3)
0
BBBBBBBBBBBBB@
hEi � hEi2PN
1
CCCCCCCCCCCCCA
/hEiC
x
3I3 + O(x7/2)
0
BBBBBBBBBBBBB@
hEi � hEi2.5PN
1
CCCCCCCCCCCCCA
/hEiC
x
7/2L7/2 + O(x4)
0
BBBBBBBBBBBBB@
hEi � hEi3PN
1
CCCCCCCCCCCCCA
/hEiC
x
4L4 + O(x9/2)
0
BBBBBBBBBBBBB@
hEi � hEi3.5PN
1
CCCCCCCCCCCCCA
/hEiC
x
9/2L9/2 + O(x5)
0
BBBBBBBBBBBBB@
hEi � hEi4PN
1
CCCCCCCCCCCCCA
/hEiC
x
5L5 + O(x11/2)
0
BBBBBBBBBBBBB@
hEi � hEi4.5PN
1
CCCCCCCCCCCCCA
/hEiC
x
11/2L11/2 + O(x6)
0
BBBBBBBBBBBBB@
hEi � hEi5PN
1
CCCCCCCCCCCCCA
/hEiC
x
6L6
0
BBBBBBBBBBBBB@
hEi � hEi5.5PN
1
CCCCCCCCCCCCCA
/hEiC
p = 1000
p = 10
20
p
µ⌫
(�),
@p
µ⌫
@t
(�),
@p
µ⌫
@r
(�)
0 ` 10
4 PN, e
10
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
New PN parameters have been confirmed with a
separate code
Flux residuals after subtracting new PN parameters
Code by Thomas Osburn
These are the main points
π�
π�π�
�π
-�
-�
-�
-�
-�
p
µ⌫
,
@p
µ⌫
@t
,
@p
µ⌫
@t
0 ` 10
4 PN, e
10
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
p
µ⌫
,
@p
µ⌫
@t
,
@p
µ⌫
@t
0 ` 10
4 PN, e
10
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
p
µ⌫
(�),
@p
µ⌫
@t
(�),
@p
µ⌫
@r
(�)
0 ` 10
4 PN, e
10
Z 2⇡
0
dt
p
d�
d�
T
r
= 2⇡M
✓
p
1� e
2
◆3/2
1 +
3 (1� e
2)
p
+O �
p
�2�
�
T
r
=
2⇡M
x
3/2+
6⇡M
(1� e
2)
px
+O �
x
1/2�
(t, r, ✓,') ! [t
p
(�), r
p
(�), ✓
p
(�),'
p
(�)]
p
tt
(t, r, ✓,') =
X
`,m
h
`m
tt
(t, r)Y
`m
(✓,')
p
tt
(�) =
X
`,m
h
`m
tt
(t
p
, r
p
)Y
`m
(✓
p
,'
p
)
p
tt
(�) =
X
`,m
⇥
¯
h
`m
tt
(t
p
, r
p
)e
�im⌦'t⇤ · ⇥Y `m
(✓
p
, 0)e
im�'(t)e
im⌦'tp⇤
p
tt
(�) =
X
`,m
¯
h
`m
tt
(t, r
p
)
¯
Y
`m
(✓
p
,'
p
)
¯
Y
`m
(✓
p
,'
p
) = Y
`m
(✓
p
,'
p
)e
�im⌦'t
h
`m,±t
=
f
2
@
r
�
r
±`m
�
h
`m,±r
=
r
2f
�
@
t
±`m
�
⌧
dE
dt
�
, New parameters, 7PN
⌧
dE
dt
�
22
= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10
�100
⌧
dE
dt
�
lm
=
X
n
!
2
64⇡
(l + 2)!
(l � 2)!
|C+lmn
|2
L4 =
1
(1� e
2)
15/2
�323105549467
3178375200
+
232597
4410
�E � 1369
126
⇡
2+
39931
294
log(2)� 47385
1568
log(3)
+
✓
�128412398137
23543520
+
4923511
2940
�E � 104549
252
⇡
2 � 343177
252
log(2) +
55105839
15680
log(3)
◆
e
2
+
✓
�981480754818517
25427001600
+
142278179
17640
�E � 1113487
504
⇡
2+
762077713
5880
log(2)� 2595297591
71680
log(3)
� 15869140625
903168
log(5)
◆
e
4
+
✓
�874590390287699
12713500800
+
318425291
35280
�E � 881501
336
⇡
2 � 90762985321
63504
log(2) +
31649037093
1003520
log(3)
+
10089048828125
16257024
log(5)
◆
e
6
+ d8e8+ d10e
10+ d12e
12+ d14e
14+ d16e
16+ d18e
18+ d20e
20+ d22e
22+ d24e
24+ d26e
26
+ d28e28
+ d30e30
+ d32e32
+ d34e34
+ d36e36
+ d38e38
+ d40e40
+ · · ·�
L7/2 = �16285⇡
504
1
(1� e
2)
7
�
1 + a2e2+ a4e
4+ a6e
6+ · · · �
⌧
dE
dt
�
=
⌧
dE
dt
�
3PN
+
32
5
⇣
µ
M
⌘2x
5
x
7/2L7/2 + x
4⇣
L4 + log(x)L4L
⌘
+ x
9/2⇣
L9/2 + log(x)L9/2L
⌘
+ x
5⇣
L5 + log(x)L5L
⌘
+ x
11/2⇣
L11/2 + log(x)L11/2L
⌘
+ x
6⇣
L6 + log(x)L6L + log
2(x)L6L2
⌘
+ x
13/2⇣
L13/2 + log(x)L13/2L
⌘
+ x
7⇣
L7 + log(x)L7L + log
2(x)L7L2
⌘
�
Questions?