10th LISA Symposium - Gainesville, Florida...Basic EOB concept: a -deformed Schwarzschild metric M m...
Transcript of 10th LISA Symposium - Gainesville, Florida...Basic EOB concept: a -deformed Schwarzschild metric M m...
10th LISA Symposium - Gainesville, Florida
Quasi-universal properties of neutron star mergers
Simone Balmelli
Physik-Institut, Universitat Zurich
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 theequation of state (EOS) of neutron stars (NS).
Is it possible (say for the advanced LIGO/Virgo network) todiscriminate between different EOS through GW detection?
In principle yes, but the late inspiral (and possibly merger) signalis needed.([Damour&al, 2012], [Del Pozzo&al, 2013], [Read&al, 2013])
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Motivation
I At present time, there is lack of knowledge about theequation of state (EOS) of neutron stars (NS).
Is it possible (say for the advanced LIGO/Virgo network) todiscriminate between different EOS through GW detection?
In principle yes, but the late inspiral (and possibly merger) signalis needed.([Damour&al, 2012], [Del Pozzo&al, 2013], [Read&al, 2013])
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Motivation
I At present time, there is lack of knowledge about theequation of state (EOS) of neutron stars (NS).
Is it possible (say for the advanced LIGO/Virgo network) todiscriminate between different EOS through GW detection?
In principle yes, but the late inspiral (and possibly merger) signalis needed.([Damour&al, 2012], [Del Pozzo&al, 2013], [Read&al, 2013])
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Which method can be used to describe neutron stardynamics close to merger?
I Numerical Relativity
I Effective One Body (EOB) approach!
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Which method can be used to describe neutron stardynamics close to merger?
I Numerical Relativity
I Effective One Body (EOB) approach!
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Which method can be used to describe neutron stardynamics close to merger?
I Numerical Relativity
I Effective One Body (EOB) approach!
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Basic EOB concept: a ν-deformed Schwarzschild metric
M
Schwarzschild
A(u) = 1 − 2u
Mtot
EOB
A(u) = P 14
[1 − 2u+ 2νu3
+
(94
3− 41
32π2)νu4 + a5νu
5
]Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Basic EOB concept: a ν-deformed Schwarzschild metric
M
m0
Schwarzschild
A(u) = 1 − 2u
Mtot
µ
EOB
A(u) = P 14
[1 − 2u+ 2νu3
+
(94
3− 41
32π2)νu4 + a5νu
5
]Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
A little bit more details...
Non-spinning EOB Hamiltonian:
HEOB = M√
1 + 2ν(Heff/µ− 1)
Heff/µ =√A(u; ν)
(1 + p2ϕu
2 + 2ν(4 − 3ν)u2p4r∗)
+ p2r∗
(See [Buonanno&Damour, 1999],[Damour&al, 2000]).
In the present work: uncalibrated model with A(u) up to 4PN,with Pade P 1
4 ; spin-orbit effects up to NNLO (∼ 3.5PN), spin-spineffects at LO (∼ 2PN), according to the lineage of [Damour, 2001]
(other versions exist!).
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
A little bit more details...
Non-spinning EOB Hamiltonian:
HEOB = M√
1 + 2ν(Heff/µ− 1)
Heff/µ =√A(u; ν)
(1 + p2ϕu
2 + 2ν(4 − 3ν)u2p4r∗)
+ p2r∗
(See [Buonanno&Damour, 1999],[Damour&al, 2000]).
In the present work: uncalibrated model with A(u) up to 4PN,with Pade P 1
4 ; spin-orbit effects up to NNLO (∼ 3.5PN), spin-spineffects at LO (∼ 2PN), according to the lineage of [Damour, 2001]
(other versions exist!).
Simone Balmelli - Universitat Zurich 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)
= −∑`≥2
κT` u2`+2(
1 + α(`)1 u+ α
(`)2 u2
)
I Tidal coupling constant
I Fractional 1PN and 2PN tidal corrections(functions of κT` and of the mass-ratio).
Simone Balmelli - Universitat Zurich 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)
= −∑`≥2
κT` u2`+2(
1 + α(`)1 u+ α
(`)2 u2
)
I Tidal coupling constant
I Fractional 1PN and 2PN tidal corrections(functions of κT` and of the mass-ratio).
Simone Balmelli - Universitat Zurich 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)
= −∑`≥2
κT` u2`+2(
1 + α(`)1 u+ α
(`)2 u2
)
I Tidal coupling constant
I Fractional 1PN and 2PN tidal corrections(functions of κT` and of the mass-ratio).
Simone Balmelli - Universitat Zurich 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)
= −∑`≥2
κT` u2`+2(
1 + α(`)1 u+ α
(`)2 u2
)
I Tidal coupling constant
I Fractional 1PN and 2PN tidal corrections(functions of κT` and of the mass-ratio).
Simone Balmelli - Universitat Zurich 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)
= −∑`≥2
κT` u2`+2(
1 + α(`)1 u+ α
(`)2 u2
)
I Tidal coupling constant
I Fractional 1PN and 2PN tidal corrections(functions of κT` and of the mass-ratio).
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
κT` = 2mB
mA
(mA
M CA
)2`+1
kA` + 2mA
mB
(mA
M CB
)2`+1
kB` .
I Compactness C =M
RI Dimensionless Love number
I EOS-dependentI measures the `-th order mass multipole moment induced by an
external, `-th order multipolar tidal field
EOS effects are essentially parametrized by the dominantcoefficient κT
2 !
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
We come to the point: tidal effects in the EOB
κT` = 2mB
mA
(mA
M CA
)2`+1
kA` + 2mA
mB
(mA
M CB
)2`+1
kB` .
I Compactness C =M
RI Dimensionless Love number
I EOS-dependentI measures the `-th order mass multipole moment induced by an
external, `-th order multipolar tidal field
EOS effects are essentially parametrized by the dominantcoefficient κT
2 !
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
The Last Stable Orbit
6 8 10 12 14r/M
0.060
0.055
0.050
0.045
0.040Heff(r,pr=
0;L
)/µ−
1
L=3.7 Mµ
L≈3.58 Mµ (LSO)
L=3.4 Mµ
ENG EOS, q=1, C=0.14
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
The Last Stable Orbit
0.02 0.04 0.06 0.08 0.10 0.12 0.14M Ωcirc
0.08
0.07
0.06
0.05
0.04
0.03
0.02Eb,circ
tidal (ENG), ν=1/4
χ1,2 =0
χ1,2 =0.2
Schw
non tidal, ν=1/4
χ1,2 =0
χ1,2 =0.2
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Results: quasi-universal relations
−0.08
−0.07
−0.06
−0.05
−0.04
EOB LSO (adiab)
NR @peak
EOB @peak
Eb
2B2HBGN1H1
ENGFPSH4
HBMPA1MS1
MS1bALF2SLy
0 100 200 300 400 500
κT2
−2−1
012
Res.
(×10
4)
0.10
0.15
0.20
0.25
EOB LSO (adiab)
EOB @peak
NR @peak
NR Fit Ref. [X]
2MΩLSO, Mω22
2B2HBGN1H1
ENGFPSH4
HBMPA1MS1
MS1bALF2SLy
0 100 200 300 400 500
κT2
−1.0−0.5
0.00.51.0
Res.
(×10
3)
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Results: quasi-universal relations
0.10
0.12
0.14
0.16
0.18
0.20
MΩ
LSO
0 100 200 300 400 500
κT2
−7.0
−6.5
−6.0
−6.5
−5.5
−5.0
Eb
LSO
(×10
2)
EOS ENG
q = 1 χ = 0
q = 1 χ = −0.1
q = 1 χ = +0.1
q = 2 χ = 0
q = 2 χ = −0.1
q = 2 χ = +0.1
20 40 60 80 100 120 1400.110.120.130.140.150.16
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Conclusions
I For any EOS, given κT2 , the EOB provides aqualitative law for determining the moment ofmerger, which is confirmed by NR.
I A single measurement of the frequency atmerger might allow to constrain the EOS.
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Conclusions
I For any EOS, given κT2 , the EOB provides aqualitative law for determining the moment ofmerger, which is confirmed by NR.
I A single measurement of the frequency atmerger might allow to constrain the EOS.
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers
Conclusions
I For any EOS, given κT2 , the EOB provides aqualitative law for determining the moment ofmerger, which is confirmed by NR.
I A single measurement of the frequency atmerger might allow to constrain the EOS.
Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers