Equation of state constraints from postmerger
gravitational-wave emission
Neutron star mergers: From gravitational waves tonucleosynthesis
16/01/2017

Andreas Bauswein(Heidelberg Institute for Theoretical Studies)with N. Stergioulas, H.-T. Janka, J. Clark

Outline: Neutron-star mergers►

Overview►

Binary mass measurements►

Dominant postmerger oscillation►

Radius measurements►

Data analysis►

Estimates of the maximum mass – collapse behavior of merger remnants►

Secondary features of the postmerger spectrum

Dynamics
Inspiral of NS binary~100 Myrs

Neutron star mergerdependent on
EoS, Mtot
ms

Prompt formation of a
BH + torus

msFormation of a differentiallyrotating massive NS
dependent on
EoS, Mtot

Rigidly rotating(supermassive) NS10-100 ms

Delayed collapse
to a BH + torus

General outcome
for 1.35-1.35 Msun binaries
Mmax with circle:
rotating NS merger
remnant
Mmax with cross:direct BH formation

42 out of 47 models lead to the formation ofa differentially rotating NS(only one accepted EoS leads to prompt collapse)

1.35-1.35 Msun, Shen EoS, rest-mass density in equatorial plane
Smooth Particle Hydrodynamics, conformally flat spatial metricBinary mass measurements

Masses from inspiral► Accurately measured “chirp mass”► Mass ratio with larger error►

i.e. q only for near-by mergers►

the closer the more accurate

Dashed red line = injected signal
Distribution function of recovered signals inblue

Rodriguez et al 2014 – injected at 100 Mpc

Total mass from chirp→ Chirp mass determines Mtot
quite well

Minimum NS mass 1.1 - 1.2 Msun (e.g. Ertl et al. 2015 or measured masses)

EoS from gravitational waves

GWs: an oversimplified picture
Two complementary approaches to infer EoS properties:●GW inspiral:strong signal - weak EoS effect(Jim's talk, Sanjay's talk; e.g. Read et al. 2013 → ~1 km @ 100 Mpc )●Postmerger oscillations:weak signal – robust strong EoS effectPostmerger gravitational-wave emission

Gravitational-wave spectrum
1.35-1.35 Msun TM1 equation of state (EoS), 20 Mpc

ringdowninspiral• Pronounced peak in the kHz range as a robust feature of all models
forming a differentially rotating NS• Characteristic GW feature: fpeak• Binary masses M1/M2 are measurable from GW inspiral signal (mostof the inspiral not covered by simulation)

Generic GW spectrum
M, EosfpeakThin line
postmerger only

Note: no uniquenomenclature in theliterature, e.g. fpeak is also
called f2 ...
• Up to two ssecondary features at lower frequencies in the postmerger spectrum(subdominan wrt to sensitivity curve) + structure at higher frequencies• Simulation: 1.35-1.35 Msun DD2 EoS, Smooth Particle Hydro, Conformal Flatness• Generic in the sense that not all secondary peaks are necessarily present• Classification scheme of postmerger emission and dynamics depending on presence/strengthof secondary peaks (Bauswein et al. 2015)

Dominant oscillation frequency• Robust feature, which occurs in all models (which don't collapse
promptly to BH)• fpeak = fundamental quadrupolar fluid mode of the remnant

Re-excitation of f-mode (l=|m|=2)
in late-time remnant (Bausweinet al. 2016)

Mode analysis at f=fpeak
Stergioulas et al. 2011

EoS dependence of fpeak

we consider relations with fixed binary masses● masses can be well measured from inspiral● in particular for distances that allow measurement of postmerger emission●

Gravitational waves – EoS survey
all 1.35-1.35 simulations

M1/M2 known frominspiralcharacterize EoS by radius ofnonrotating NS with 1.35 Msun

Pure TOV/EoS property => Radius measurement via fpeak

Important: Simulations for the same binary system, just with varied EoS
Bauswein et al. 2012

Gravitational waves – EoS survey
all 1.35-1.35 simulations

M1/M2 known frominspiralcharacterize EoS by radius ofnonrotating NS with 1.6 Msun
Bauswein et al. 2012

Pure TOV/EoS property => Radius measurement via fpeak

Important: Simulations for the same binary system, just with varied EoS
Note: R of 1.6 Msun NS scales with fpeak from 1.35-1.35
Msun mergers (density regimes comparable)

Remarks►

Similar relations for asymmetric binaries and other binary masses►(mass will be measured by inspiral)►Initial intrinsic NS rotation unimportant►

Agreement with different groups (Kyoto, Caltech, Frankfurt, Parma, e.g. Hotokezaka et
al 2013, Bernuzzi et al. 2015, Takami et al. 2015, De Pietri et al. 2015, Foucart et al.
2016)→ No impact on fpeak

Bauswein et al. 2016

Strategy►

Measure binary masses from inspiral►Construct fpeak – R relation for fixed binary mass►

Measure fpeak from postmerger GW signal►

Obtain radius by inverting fpeak – R relation

Binary mass variations

Different total binary masses(symmetric)

Fixed chirp mass (incl.
Asymmertic binaries)
Bauswein et al. 2016

Detectability

Unmodeled searches: Measuring thedominant GW frequency

Model waveforms hidden in
rescaled LIGO noise
Peak frequency recovered withburst search analysis
Error ~ 10 Hz
For signals within ~10-25 Mpc=> for near-by event radius
measurable with high precision(~0.01-1/yr)
Proof-of-principle study→ improvements likely

Clark et al. 2014

Unmodeled search – less sensitive

Discrimination between prompt collapse and delay/no collapse possible
Shen 1.35-1.35; Clark et al. 2014

Principal Component Analysis

Only first component
Excluding the reconstructed waveform from catalogue

Clark et al 2016

PCA - prospects
Note:● rate unclear, could bemuch higher● Conservative estimate –numerical damping andfinite duration→underestimation of signal
strength
Systematics error:● Intrinsic scatter in relation● Uncertainties in simulations
Better error for closer systems
Clark et al. 2016

→ improvements possible

Collapse behavior and the maximum mass of(non-rotating) neutron stars
Key quantity: Threshold binary mass for prompt collapse

Note: Collapse behavior important for nucleosynthesis,kilonovae and short gamma-ray bursts !!!

Collapse behavior
Total binary mass Mtot

Prompt collapse to BH
InspiralNo or delayed collapse to BH

Threshold binary
mass Mthres

EoS dependent !!! - somehow Mmax should play a role

Simulations

M thres
k=M max

from simulations
TOV property of employed EoS

Simulations with manydifferent binary masses

Key quantity: Threshold binary mass Mthres for prompt BH collapse

Fractional increase of Mmax

Mthres = k * Mmax
with k = k(Cmax)
Cmax = G Mmax / (c2 Rmax)(compactness of TOV
maximum-mass configuration)=> Mthres = Mthres(Mmax,Rmax)
Bauswein et al. 2013

M thres
k=M max

From simulations with different Mtot
TOV property of employed EoS

Fit:e.g. from fpeak measurement

Mmax - Method 1►

Measure several NS mergers – check if postmerger GW emission present→ Mthres estimate►(We ~ know radii at that point)►

Invert fit→ Mmax►

Note: already a single/few measurement could
provide interesting constraints !!!

Mmax – Method 2►

Possibly there are not many binaries with Mtot ~ Mthres►

May be hard to measure Mthres►

Method 2: extrapolate from observations as “moderate” binary masses►

e.g. two events: one with 1.25-1.25 Msun and one with 1.4-1.4 Msun

fpeak dependence on total binary mass(every single line
corresponds to a
specific EoS→ only one line canbe the true EoS)

Dominant GW frequency monotone function of Mtot
Threshold to prompt BH collapse shows a clear dependence on Mtot
(dashed line)Threshold to prompt BH collapse

Extrapolation procedure
Details in Bauswein
et al. 2014

Two fpeak measurements at different Mtot yield threshold massand “threshold frequency” !!!

Extrapolation procedure

fthresMthresTwo fpeak measurements at different Mtot yield threshold massand “threshold frequency” !!!

Rmax determination via extrapolation

Threshold frequency fthres yields a good estimate of the radiusof the TOV maximum mass configuration (a few 100 meters)

Maximum mass via extrapolation

Mmax within 0.1 Msun, Rmax within a few 100 m(from fpeak detections at common Mtot)

from two measurements of fpeak at moderate Mtot

(final error will depend on EoS and extact systems measured)
Note: Mthres may also be constrained from prompt collapse directlyMaximum density via extrapolation

Maximum density of non-rotating NS within 10 per centSecondary GW peaks

Generic GW spectrum
f2-0

fspiralfpeak• Orbital motion of antipodal bulges generate peak at fspiral• Coupling between quasi-radial and quadrupolar mode
Bauswein & Stergioulas 2015

pirafpeak always

fsf2-0higl lohw

MM

tot

tot

Survey of GW spectra• Considering different models (EoS, Mtot): 3 types of spectra depending on
presence of secondary features (dominant fpeak is always present)• Secondary frequencies depend on EoS (similar as fpeak)
Bauswein & Stergioulas 2015

Survey of GW spectra

Type I

Type II

Type III

LS220, DD2, NL3 EoS all with Mtot = 2.7 Msun → consider Mtot relative Mthres=> Depending on binary model (EoS, M1/2) either one or the other or both features arepresent / dominant (if you measure a secondary peak you should always think whether
it is f2-0 or fspiral)

Classification scheme
●●

●

●

●

●

Type I: 2-0 feature dominates, fspiral hardly visible, radial mode stronglyexcited, observed for soft EoS, relatively high M tot
Type II: both secondary features have comparable strength, clearly
distinguishable, moderate binary masses
Type III: fspiral dominates, f2-0 hardly visible, found for stiff EoS,
relatively low binary masses, (central lapse, GW amplitude, rhomax
show low-frequency modulation in addition to radial oscillation)
Different types show also different dynamical behavior, e.g. in central lapse, maximum density,GW amplitude, ….
High mass / low mass relative to threshold binary mass for prompt BH collapse (→ EoSdependent)
Continuous transition between different types: a given EoS shows all types depending onMtot: Type III for low Mtot → Type I towards Mthres
Bauswein & Stergioulas 2015

Classification schemeType of M1-M2 merger indicate at Mtot/2 = M1

Bauswein et al. 2015

(Continuous transition between types → tentative association)
For Mtot = 2.7 Msun all Types are possible depending on EoSClassification intuitive: merger dynamics affected by compactness

Dependencies of secondary frequencies
1.35-1.35 Msun

Clark et al. 2016

→ secondary frequencies are essentially given by dominant frequency

Conclusions►

NS radii scale tightly with dominant postmerger GW frequency►

GW data analysis → radii measurable with precision of a few 100 m►a single measurement of only the dominant frequency is sufficient►

Collapse behavior determined by EoS → maximum mass Mmax of non-rotating NSs- directly determine Mthres by several merger events- extrapolation method: 2 events with “moderate” binary masses►Radius Rmax of maximum-mass TOV star►

Maximum density in NSs►

Secondary peaks generated by distinct mechanisms, EoS dependent, classification
scheme based on presence of secondary features