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

Which method can be used to describe neutron stardynamics close to merger?

I Numerical Relativity

I Effective One Body (EOB) approach!

Basic EOB concept: a ν-deformed Schwarzschild metric

Schwarzschild

A(u) = 1 − 2u

A(u) = P 14

[1 − 2u+ 2νu3

3− 41

32π2)νu4 + a5νu

]Simone Balmelli - Universitat Zurich Quasi-universal properties of neutron star mergers

Basic EOB concept: a ν-deformed Schwarzschild metric

Schwarzschild

A(u) = 1 − 2u

A(u) = P 14

[1 − 2u+ 2νu3

3− 41

32π2)νu4 + a5νu

]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!).

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!).

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).

A(u) = A0(u) + AT (u)

= −∑`≥2

κT` u2`+2(

1 + α(`)1 u+ α

(`)2 u2

A(u) = A0(u) + AT (u)

= −∑`≥2

κT` u2`+2(

1 + α(`)1 u+ α

(`)2 u2

A(u) = A0(u) + AT (u)

= −∑`≥2

κT` u2`+2(

1 + α(`)1 u+ α

(`)2 u2

A(u) = A0(u) + AT (u)

= −∑`≥2

κT` u2`+2(

1 + α(`)1 u+ α

(`)2 u2

κT` = 2mB

kA` + 2mA

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

κT` = 2mB

kA` + 2mA

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

The Last Stable Orbit

6 8 10 12 14r/M

0.040Heff(r,pr=

)/µ−

L=3.7 Mµ

L≈3.58 Mµ (LSO)

L=3.4 Mµ

ENG EOS, q=1, C=0.14

The Last Stable Orbit

0.02 0.04 0.06 0.08 0.10 0.12 0.14M Ωcirc

0.02Eb,circ

tidal (ENG), ν=1/4

χ1,2 =0

χ1,2 =0.2

non tidal, ν=1/4

χ1,2 =0

χ1,2 =0.2

Results: quasi-universal relations

−0.08

−0.07

−0.06

−0.05

−0.04

EOB LSO (adiab)

NR @peak

EOB @peak

2B2HBGN1H1

ENGFPSH4

HBMPA1MS1

MS1bALF2SLy

0 100 200 300 400 500

−2−1

EOB LSO (adiab)

EOB @peak

NR @peak

NR Fit Ref. [X]

2MΩLSO, Mω22

2B2HBGN1H1

ENGFPSH4

HBMPA1MS1

MS1bALF2SLy

0 100 200 300 400 500

−1.0−0.5

0.00.51.0

Results: quasi-universal relations

0 100 200 300 400 500

−7.0

−6.5

−6.0

−6.5

−5.5

−5.0

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

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.

Conclusions

10th lisa symposium - gainesville, florida...basic eob concept: a -deformed schwarzschild metric m m...

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