Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Spin Alignment Effects inStellar Mass Black Hole Binaries
Davide GerosaCo-authors: Emanuele Berti, Michael Kesden,
Ulrich Sperhake, Richard O’Shaughnessy
University of MississippiOxford, MS
September 29, 2012
22nd Midwest Relativity MeetingChicago, IL
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Contents
1. Post-Newtonian Spin-Orbit Resonances
2. Stellar-Mass Black Hole Binary Formation
3. Results
4. Future Developments
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Post-Newtonian Evolution
Black hole binary inspiral:
• Large separation R > 1000M:interactions with the astrophysical environment.
• Post-Newtonian approximation:solve the Einstein field equations by a series in v/c .
• Small separation R < 10M:numerical relativity.
Can the PN evolution alter spin orientation?
If the evolution brings the spin parameter to be clustered in certainregions we could place more templates in these regions, increasing theefficiency of possible detections.
Evolution from R=1000M to R=10M: what happens to the spins?
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Equations of MotionKidder 1995; Arun et al. 2011; Kesden, Sperhake, and Berti 2010a
Spin precessiondS1
dt= Ω1 × S1
dS2
dt= Ω2 × S2
Ω1 = v5 3
4+η
2−
3
4
δm
M
L +1
2
v6
M2
[S2 − 3
(L · S2
)L]−
3
2
v6
M2
m2
m1
(L · S1
)L
Angular momentum conservation and radiation reactiond L
dt= −
ηv
M2
S1 + S2
dv
dt=
32
5
η
Mv9
1 + v2 − 743
336−
11
4η
+ v34π − ∑
i=1,2χi (Si · L)
113
12
m2i
M2+
25
4η
+ v4
3410318144
+13661
2016η +
59
18η2+
+721
48ηχ1χ2(S1 · L)(S2 · L) −
247
48ηχ1χ2(S1 S2) +
∑i=1,2
5
2χ2i
mi
M
2 (3(Si · L)2 − 1
)+
+∑
i=1,2
1
96χ2i
mi
M
2 (7 − (Si · L)2
) + v5− 4159
672−
189
8η
π + v6
16447322263139708800
+16
3π2 −
1712
105
(γE + ln 4v
)+
+
− 56198689
217728+
451
48π2 η +
541
896η2 −
5603
2592η3 + v7
π
− 4415
4032+
358675
6048η +
91495
1512η2 + O(v8)
Results robust to the addition of further PN terms [Favata, in prep.]Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Resonances
θ1
S1
LN
ey
ex∆φ
θ2
S2
Schnittman 2004
Family of equilibrium solutions in which thetwo BH spins and the orbital angularmomentum are coplanar, precessing jointlyabout the total angular momentum
• Fixing L, i.e. at a particular point in spaceand time during the inspiral
• ∆Φ = 0, 180
• θ1 and θ2 solution of the PN equations
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Resonances
θ1
S1
LN
ey
ex∆φ
θ2
S2
Schnittman 2004
Family of equilibrium solutions in which thetwo BH spins and the orbital angularmomentum are coplanar, precessing jointlyabout the total angular momentum
• Fixing L, i.e. at a particular point in spaceand time during the inspiral
• ∆Φ = 0, 180
• θ1 and θ2 solution of the PN equations
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Resonances q = 9/11, χ1 = 1, χ2 = 1
From r = 1000M to r = 10M
Kesden, Sperhake, and Berti 2010a
Two Resonances
• θ1 < θ2 → ∆Φ = 0
• θ2 < θ1 → ∆Φ = 180
Gravitational radiation: emissionof energy. The system evolvestowards θ1 ' θ2 along the red curves
Locking: if the system gets close toa resonance, it can get trapped!
rlock =
(1 + q2
1− q2
)2
M
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Resonances q = 9/11, χ1 = 1, χ2 = 1
From r = 1000M to r = 10M
Kesden, Sperhake, and Berti 2010a
Two Resonances
• θ1 < θ2 → ∆Φ = 0
• θ2 < θ1 → ∆Φ = 180
Gravitational radiation: emissionof energy. The system evolvestowards θ1 ' θ2 along the red curves
Locking: if the system gets close toa resonance, it can get trapped!
rlock =
(1 + q2
1− q2
)2
M
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Isotropic sample q = 9/11, χ1 = 1, χ2 = 1
From left to right and from top to bottom: Initial conditions, r = 1000M,
r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M
cos θ1 vs. cos θ2 ∆Φ vs. cos θ12
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Compact binary formation: looking for initial conditions
Standard population synthesis model by Belczynski et al. 2008
• Binary stars: spins are initially aligned (tidal interactions)
• Stellar evolution, tracking those that produce BH-BH.
• SN explosions: each new compact object receives a kick
• Kick’s magnitudes are taken from the observed pulsar propermotions distribution and the direction is assumed isotropic
• Asymmetric mass ejection: fallback material
• Kicks change the orbital plane, not the direction of the spins θ1 ∼ θ2Is there any realignment due to mass transfers and tidal interactionsbetween the two SN?
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Simulation Setup
• Mass ratio q = 9/11 ' 0.81:• It is the value used in all the previous works (comparison)• Median and average of the catalog by Dominik et al. 2012 are ' 0.83
• Maximally spinning BHs: χ1 = χ2 = 1
• Misalignment of 10 with a dispersion of 3
• ∆Φ free (initial phase of the precession)
Phenomenological scenarios
• 10/10: Both the BHs are tiltedθ1 ∈ [10 − 3, 10 + 3] θ2 ∈ [10 − 3, 10 + 3]
• 10/0: Secondary BH realigned*θ1 ∈ [10 − 3, 10 + 3] θ2 ∈ [0, 3]
• 0/10: Primary BH realignedθ1 ∈ [0, 3] θ2 ∈ [10 − 3, 10 + 3]
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
10/10 sample q = 9/11, χ1 = 1, χ2 = 1
From left to right and from top to bottom: Initial conditions, r = 1000M,
r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M
cos θ1 vs. cos θ2 ∆Φ vs. cos θ12
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
10/0 sample 0/10 sample
From left to right and from top to bottom: Initial conditions, r = 1000M,
r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M
∆Φ vs. cos θ12 ∆Φ vs. cos θ12
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Towards an Astrophysical Model
Results
• Locking strongly depends on initial conditions
• Tidal interactions and mass transfer events are crucial
Outlook:
• Analytical Model of the kick, using conservation laws
• Generalization of Kalogera 2000 to elliptic orbits and double kicks
• Priors: progenitor masses, initial separation, eccentricity
• Tidal interactions and mass transfers
⇒ Initial tilt angle distributions
• PN evolution and resonant effects
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Acknowledgements
E. Berti,M. Kesden,U. Sperhake,R. O’Shaughnessy,M. Favata,G. Lodato.
Thanks
Davide [email protected]
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
General Picture
BH - BH Binary main parameters:
• Source position: right ascension, declination, distance
• Orbital plane: two angles (orientation and polarization angle)
• Phase at coalescence, time of coalescence
⇒ extrinsic parameters: affect the amplitude of the signal, but not its form
• Masses: (m1,m2) or (M, q)
⇒ 9 parameters for non-spinning binaries (circular orbit)
• Spin vectors: components, or magnitude and two angles for each BH
⇒ 15 parameters for spinning binaries (circular orbit)
⇒ intrinsic parameters: affect the evolution of amplitude and phase in time
• Orbit: eccentricity and orientation of semi-major axis (two angles)
⇒ 18 parameters for spinning binaries on a general orbit
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Black Hole Spins
Do we need all of these parameters?
The orbits circularize on a timescale shorter than the inspiral timescale(Peters 1964; Peters and Mathews 1963)
Are astrophysical black holes spinning?
For Kerr black holes, the spin angular momentum is given by
J = χM2G
c
with 0 ≤ χ ≤ 1
We cannot avoid the use of the 6 spins parameters but we can (maybe)find out something about spin alignment.
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Black Hole Spins
Measurements indicate that many BHs could be rapidly spinning
• Fe K-α line profile
• Continuos X-Ray emission of the accretion disc
0.5 1 1.5
Line profile
GraviGeneral relativity
Tran
Beami
Special relativity
Newtonian
r t tr t
❳
❳
❯
❳
❳
rrtr r qt P t t♦❱ ❲rtt r rt r t ❨ ❩❱
Fabian et al. 2000
McClintock et al. 2011
We cannot avoid the use of the 6 spins parameters but we can (maybe)find out something about spin alignment.
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Equations of MotionKidder 1995; Arun et al. 2009; Arun et al. 2011
Spin PrecessiondS1
dt= Ω1 × S1
dS2
dt= Ω2 × S2
Ω1 = v5(
3
4+η
2−
3
4
δm
M
)L Ω2 = v5
(3
4+η
2+
3
4
δm
M
)L
Angular Momentum Conservation and Radiation Reaction
d L
dt= −
ηv
M2
S1 + S2
dv
dt=
32
5
η
Mv9
1 + v2
[−
743
336−
11
4η
]+ v3
[4π −
∑i=1,2
χi (Si · L)
(113
12
m2i
M2+
25
4η
)]+ O(v4)
where M = m1 + m2, η =m1m2M2 , δm = m1 − m2.
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Equations of MotionKesden, Sperhake, and Berti 2010a; Kesden, Sperhake, and Berti 2010b; Berti, Kesden, and Sperhake 2012
Spin PrecessiondS1
dt= Ω1 × S1
dS2
dt= Ω2 × S2
Ω1 = v5 3
4+η
2−
3
4
δm
M
L +1
2
v6
M2
[S2 − 3
(L · S2
)L]−
3
2
v6
M2
m2
m1
(L · S1
)L
Angular Momentum Conservation and Radiation Reaction
d L
dt= −
ηv
M2
S1 + S2
dv
dt=
32
5
η
Mv9
1 + v2 − 743
336−
11
4η
+ v34π − ∑
i=1,2χi (Si · L)
113
12
m2i
M2+
25
4η
+ v4
3410318144
+13661
2016η +
59
18η2+
+721
48ηχ1χ2(S1 · L)(S2 · L) −
247
48ηχ1χ2(S1 S2) +
∑i=1,2
5
2χ2i
mi
M
2 (3(Si · L)2 − 1
)
+∑
i=1,2
1
96χ2i
mi
M
2 (7 − (Si · L)2
) + v5− 4159
672−
189
8η
π + v6
16447322263139708800
+16
3π2 −
1712
105
(γE + ln 4v
)+
+
− 56198689
217728+
451
48π2 η +
541
896η2 −
5603
2592η3 + v7
π
− 4415
4032+
358675
6048η +
91495
1512η2 + O(v8)
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
PN Equations of MotionNew PN terms computed by M.Favata
Spin PrecessiondS1
dt= Ω1 × S1
dS2
dt= Ω2 × S2
Ω1 = v5 3
4+η
2−
3
4
δm
M
L +1
2
v6
M2
[S2 − 3
(L · S2
)L]−
3
2
v6
M2
m2
m1
(L · S1
)L + v7
9
16+
5η
4−η2
24+δm
M
− 9
16+
5η
8
L
Angular Momentum Conservation and Radiation Reactiond L
dt= −
ηv
M2
1 + v2 3
2+η
6
−1
S1 + S2 −2η
Mv7
η +
m2
M
21 +
3
16
m2
m1
(L × S1
)−
(1 ↔ 2
)dv
dt=
32
5
η
Mv9
1 + v2 − 743
336−
11
4η
+ v34π − ∑
i=1,2χi (Si · L)
113
12
m2i
M2+
25
4η
+ v4
3410318144
+13661
2016η +
59
18η2+
+721
48ηχ1χ2(S1 · L)(S2 · L) −
247
48ηχ1χ2(S1 S2) +
∑i=1,2
5
2χ2i
mi
M
2 (3(Si · L)2 − 1
)+
∑i=1,2
1
96χ2i
mi
M
2 (7 − (Si · L)2
) +
+ v5− 4159
672−
189
8η
π +∑
i=1,2χi (Si · L)
−21219
1008
m2i
M2+
1159
24η
m2i
M2−
809
84η +
281
8η
+
−1
4
∑i=1,2
mi
M
3 χi (Si · L)(1 + 3χ2i )
1 −15
8χ2i
1 − (Si · L)2
1 + 3χ2i
+ v6
16447322263139708800
+16
3π2 −
1712
105
(γE + ln 4v
)+
+
− 56198689
217728+
451
48π2 η +
541
896η2 −
5603
2592η3 + v7
π
− 4415
4032+
358675
6048η +
91495
1512η2 + O(v8)
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Correlation between θ12 and ∆Φ
• θ12: angle between the two spins
• ∆Φ: angle between their projections on the orbital plane
cos θ12 = sin θ1 sin θ2 cos ∆Φ + cos θ1 cos θ2
Expected correlation: integrate over θ1 and θ2 providing initialdistributions f1 = f1(cos θ1) and f2 = f2(cos θ2)
< cos θ12 >=
∫sin θ1 f1 d cos θ1∫
f1 d cos θ1
∫sin θ2 f2 d cos θ2∫
f2 d cos θ2cos ∆Φ+
+
∫cos θ1 f1 d cos θ1∫
f1 d cos θ1
∫cos θ2 f2 d cos θ2∫
f2 d cos θ2
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Constant of Motion
Effective One-Body problem (Damour 2001)
S0 = (1 + q)S1 + (1 + q−1)S2
At 2PN order (v/c)4
S0 · L
is a constant of motion (Racine 2008)
• (cos θ1, cos θ2) plane: propagation with slopes − 1q
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
10/0 sample q = 9/11, χ1 = 1, χ2 = 1
From left to right and from top to bottom: Initial conditions, r = 1000M,
r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M
cos θ1 vs. cos θ2 ∆Φ vs. cos θ12
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
0/10 sample q = 9/11, χ1 = 1, χ2 = 1
From left to right and from top to bottom: Initial conditions, r = 1000M,
r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M
cos θ1 vs. cos θ2 ∆Φ vs. cos θ12
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Average < |∆Φ| >Is
otr
op
ic
10
/1
0
0/
10
10
/0
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Average < cos θ12 >
Iso
tro
pic
10
/1
0
0/
10
10
/0
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
SN explosion on Eccentric Orbits
Elliptic orbit M0, a0, e0.
r0 =a0(1− e20 )
1 + e0 cosψ0
The explosion happens with true anomaly ψ0.Kick: mangnitude vk , direction θ, φ.
Conservation laws −→ new orbit parameters M1, a1, e1
α =a1a0
β =M1
M0uk =
vk√GM0/a0
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
SN explosion on Eccentric Orbits
Conservation of energy: semi-major axis
a1a0
= β
[2 (β − 1)
(1 + e0 cosψ0
1− e20
)+1−u2
k−2uk
(2
1 + e0 cosψ0
1− e20− 1
)1/2
cos θ
]−1
Conservation of angular momentum: eccentricity
(1− e21
)=
1
αβ
[(√1− e20 +
1− e201 + e0 cosψ0
uk cos θ
)2
+
+
(1− e20
1 + e0 cosψ0uk sin θ cosφ
)2].
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
SN explosion on Eccentric Orbits
Tilt Angle (single kick)
cos γ =L0 · L1
|L0||L1|=
=
(√1− e20 +
1−e201+e0 cosψ0
uk cos θ)
[(√1− e20 +
1−e201+e0 cosψ0
uk cos θ)2
+(
1−e201+e0 cosψ0
uk sin θ cosφ)2 ]1/2 .
• Generalization of the expression in Kalogera 2000 for e0 6= 0
• Double kick: using the same expressions, updating the system ofreference
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
Preliminary: nothing happens between the two SN
• No mass loss: ms1 = m1 = 11M, ms2 = m2 = 9M• a0 = 10R, e0 = 0
0 50 100 1500
0.005
0.01
0.015
tilt angle
0 50 100 1500
0.2
0.4
0.6
0.8
1
tilt angle
Tilt angle distributions (double kick)
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries
Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments
From left to right and from top to bottom: Initial conditions, r = 1000M,
r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M
cos θ1 vs. cos θ2 ∆Φ vs. cos θ12
Davide Gerosa University of Mississippi
Spin Alignment Effects in Stellar Mass Black Hole Binaries