10th LISA Symposium - Gainesville, Florida
Quasi-universal properties of neutron star mergers
Simone Balmelli
Physik-Institut, Universität Zürich
May 22, 2014
Paper by: Sebastiano Bernuzzi, Alessandro Nagar, Simone Balmelli, Tim Dietrich,
Maximiliano Ujevic
Motivation
I
At present time, there is lack of knowledge about the
equation of state (EOS) of neutron stars (NS).
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Motivation
I
At present time, there is lack of knowledge about the
equation of state (EOS) of neutron stars (NS).
Is it possible (say for the advanced LIGO/Virgo network) to
discriminate between different EOS through GW detection?
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Motivation
I
At present time, there is lack of knowledge about the
equation of state (EOS) of neutron stars (NS).
Is it possible (say for the advanced LIGO/Virgo network) to
discriminate between different EOS through GW detection?
In principle yes, but the late inspiral (and possibly merger) signal
is needed.
([Damour&al, 2012], [Del Pozzo&al, 2013], [Read&al, 2013])
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Which method can be used to describe neutron star
dynamics close to merger?
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Which method can be used to describe neutron star
dynamics close to merger?
I
Numerical Relativity
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Which method can be used to describe neutron star
dynamics close to merger?
I
Numerical Relativity
I
Effective One Body (EOB) approach!
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Basic EOB concept: a ν-deformed Schwarzschild metric
Schwarzschild
M
A(u) = 1 − 2u
Simone Balmelli - Universität Zürich
EOB
Mtot
A(u) = P41 1 − 2u + 2νu3
94 41 2
4
5
+
− π νu + a5 νu
3
32
Quasi-universal properties of neutron star mergers
Basic EOB concept: a ν-deformed Schwarzschild metric
Schwarzschild
m0
M
A(u) = 1 − 2u
Simone Balmelli - Universität Zürich
EOB
µ
Mtot
A(u) = P41 1 − 2u + 2νu3
94 41 2
4
5
+
− π νu + a5 νu
3
32
Quasi-universal properties of neutron star mergers
A little bit more details...
Non-spinning EOB Hamiltonian:
q
HEOB = M 1 + 2ν(Hef f /µ − 1)
q
Hef f /µ = A(u; ν) 1 + p2ϕ u2 + 2ν(4 − 3ν)u2 p4r∗ + p2r∗
(See [Buonanno&Damour, 1999],[Damour&al, 2000]).
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
A little bit more details...
Non-spinning EOB Hamiltonian:
q
HEOB = M 1 + 2ν(Hef f /µ − 1)
q
Hef f /µ = A(u; ν) 1 + p2ϕ u2 + 2ν(4 − 3ν)u2 p4r∗ + p2r∗
(See [Buonanno&Damour, 1999],[Damour&al, 2000]).
In the present work: uncalibrated model with A(u) up to 4PN,
with Padé P41 ; spin-orbit effects up to NNLO (∼ 3.5PN), spin-spin
effects at LO (∼ 2PN), according to the lineage of [Damour, 2001]
(other versions exist!).
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
From [Damour&Nagar, 2010]:
A(u) = A0 (u) + AT (u)
=−
X
(`)
(`)
κT` u2`+2 1 + ᾱ1 u + ᾱ2 u2
`≥2
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
From [Damour&Nagar, 2010]:
I
Point-particle EOB radial potential
A(u) = A0 (u) + AT (u)
=−
X
(`)
(`)
κT` u2`+2 1 + ᾱ1 u + ᾱ2 u2
`≥2
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
From [Damour&Nagar, 2010]:
I
Point-particle EOB radial potential
A(u) = A0 (u) + AT (u)
=−
I
X
(`)
(`)
κT` u2`+2 1 + ᾱ1 u + ᾱ2 u2
`≥2
Tidal coupling constant
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
From [Damour&Nagar, 2010]:
I
Point-particle EOB radial potential
A(u) = A0 (u) + AT (u)
=−
X
(`)
(`)
κT` u2`+2 1 + ᾱ1 u + ᾱ2 u2
`≥2
I
Tidal coupling constant
I
Fractional 1PN and 2PN tidal corrections
(functions of κT` and of the mass-ratio).
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
κT`
I
I
mB
=2
mA
mA
M CA
!2`+1
k`A
mA
+2
mB
mA
M CB
!2`+1
k`B .
M
R
Dimensionless Love number
Compactness C =
I
I
EOS-dependent
measures the `-th order mass multipole moment induced by an
external, `-th order multipolar tidal field
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
κT`
I
I
mB
=2
mA
mA
M CA
!2`+1
k`A
mA
+2
mB
mA
M CB
!2`+1
k`B .
M
R
Dimensionless Love number
Compactness C =
I
I
EOS-dependent
measures the `-th order mass multipole moment induced by an
external, `-th order multipolar tidal field
EOS effects are essentially parametrized by the dominant
coefficient κT2 !
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
The Last Stable Orbit
Heff(r,pr =0;L)/µ−1
0.040
0.045
L =3.7 Mµ
0.050
L ≈ 3.58 Mµ (LSO)
L =3.4 Mµ
0.055
0.060
ENG EOS, q =1, C =0.14
6
8
Simone Balmelli - Universität Zürich
10
r/M
12
14
Quasi-universal properties of neutron star mergers
The Last Stable Orbit
0.02
tidal (ENG), ν =1/4
0.03
Schw
χ1,2 =0
χ1,2 =0.2
Eb,circ
0.04
0.05
0.06
χ1,2 =0
0.07
0.08
non tidal, ν =1/4
0.02
0.04
0.06
Simone Balmelli - Universität Zürich
0.08
M Ωcirc
0.10
χ1,2 =0.2
0.12
0.14
Quasi-universal properties of neutron star mergers
Results: quasi-universal relations
2M ΩLSO , M ω22
Eb
−0.04
2B
2H
BGN1H1
ENG
FPS
H4
HB
MPA1
MS1
MS1b
ALF2
SLy
2B
2H
BGN1H1
0.25
ENG
FPS
H4
HB
MPA1
MS1
MS1b
ALF2
SLy
EOB LSO (adiab)
−0.05
EOB @peak
0.20
−0.06
NR @peak
EOB LSO (adiab)
NR Fit Ref. [X]
EOB @peak
0
100
200
0.10
300
400
500
κT2
Simone Balmelli - Universität Zürich
Res. (×103 )
Res. (×104 )
−0.08
2
1
0
−1
−2
0.15
NR @peak
−0.07
1.0
0.5
0.0
−0.5
−1.0
0
100
200
300
400
κT2
Quasi-universal properties of neutron star mergers
500
Results: quasi-universal relations
0.20
0.16
0.15
0.14
0.13
0.12
0.11
20
M ΩLSO
0.18
0.16
0.14
40
60
80
100
120
140
0.12
Eb LSO (×102 )
0.10
−5.0
−5.5
−6.0
q
q
q
q
q
q
EOS ENG
−6.5
−7.0
0
100
200
300
=1χ=0
= 1 χ = −0.1
= 1 χ = +0.1
=2χ=0
= 2 χ = −0.1
= 2 χ = +0.1
400
500
κT2
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Conclusions
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Conclusions
I
For any EOS, given κT2 , the EOB provides a
qualitative law for determining the moment of
merger, which is confirmed by NR.
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers
Conclusions
I
I
For any EOS, given κT2 , the EOB provides a
qualitative law for determining the moment of
merger, which is confirmed by NR.
A single measurement of the frequency at
merger might allow to constrain the EOS.
Simone Balmelli - Universität Zürich
Quasi-universal properties of neutron star mergers