Why study the late inspiral?

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2 Why study the late inspiral? 2 point-like inspiral, hydrodynamical inspiral and merger. observation of the hydrodynamical inspiral or the merger phase constraints on EOS the late inspiral the initial conditions for the merger phase... f GW Hz, importance of the crust 0.22? h(t) PN Effective One Body computation [Buonanno & Damour, PRD 62, (2000)] inspiral + plunge merger + ring down naive LSO r LSO j LSO inspiral Ε LSO ω LSO plunge ν + merger = 1/ t/m QNM ringdown Buonanno & Damour, PRD 62, (2000)

3 Defining an evolutionary sequence of binary NS 3 The inspiral phase can be studied by constructing quasi-equilibrium sequences of close NS binaries in Isenberg-Wilson-Mathews formulation. The quasi-stationary evolution of binary NS is modelled by computing a sequence of configurations with decreasing separation and fixed total baryon number The fluid in NS has zero vorticity in the inertial frame (irrotational case) The end of quasi-equilibrium : the last stable orbit ISCO (one of observable parameters by GW detectors): orbital instability or mass-shed limit The numerical method : A multi-domaines 3D spectral method Surface fitted cordinates Codes C++ Lorene ( Surface fitted cordinates

4 4 E bind [M sol ] SQS MIT irrot. SQS MIT corot. Polytrope γ = 2.5 irrot. Polytrope γ = 2.5 corot. 3PN irrot. (Blanchet 2002) 3PN corot. (Blanchet 2002) 3PN irrot. (Damour et al. 2002) 3PN corot. (Damout et al. 2002) f GW [Hz] [Limousin, Gondek-Rosinska, Gourgoulhon (2005)]

5 What is the EOS of matter at supranuclesr densities? 5 SS hypothesis: A.R. Bodmer (1971): the absolute ground state of nuclear matter may be a state of deconfined up, down and strange quarks E. Witten (1984) reformulated this idea, and contemplated the possibility that neutron stars are in fact strange quark stars. quark-hybrid star N+e N+e+n traditional neutron star hyperon star s u p e r c o n,p,e, µ n d u c t i n g p neutron star with pion condensate absolutely stable strange quark matter µ u d s Σ,Λ,Ξ, u,d,s quarks H n,p,e, µ K π r o t n superfluid n o s crust Fe 10 6 g/cm g/cm g/cm 3 m s strange star R ~ 10 km M ~ 1.4 M nucleon star

6 Static neutron stars and strange stars 6 First numerical models computed by Haensel, Zdunik & Schaeffer [A&A 160, 121 (1986)] and Alcock, Fahri & Olinto [ApJ 310, 261 (1986)] by integration of the Tolman-Oppenheimer-Volkoff equations with MIT bag-model EOS. Basic features: finite density at the surface (zero pressure) for small mass (weak gravity): almost constant density profile ε 4B

7 Static neutron stars and strange quark stars AkmalPR SQSB40 Neutron Stars EOS - solid lines M [M o ] SQSB56 BPAL12 SQSB60 GlendNH M ȯ AkmalPR: neutrons, protons, electrons and muons described by the A18+dv+UIX* model of Akmal et al BPAL12: n, p, e, muons Bombaci et al 1995 GlendNH3: nucleons + hyperons Glendenning R [km] Strange Quark Stars MIT Bag model - dashed lines The MIT Bag model parameters: B - the bag constant, density at the stellar surface; m s - the strange quark mass; α - the coupling constant; SQSB40 - B=40 MeV/fm 3 m s =100 MeV,α = 0.6; SQSB56 B=56 MeV/fm 3 m s =200 MeV,α = 0.2; SQSB60 B=60 MeV/fm 3 m s =0,α = 0

8 Isocontours of baryon density for irrotational binary neutron stars described by realistic EOS 8

9 The last orbits of binary strange stars with masses 1.35M E bind [M sol ] SQSB40 SQSB60 SQSB56 GlendNH3 BPAL12 AkmalPR 3PN (Blanchet 2002) E bind - E bind 3PN [Msol ] 0,003 0,002 0,001 SQSB40 SQSB56 SQSB60 DSQS f GW [Hz] 0 0,6 0,8 1 1,2 1,4 1,6 1,8 2 2,2 f GW [khz] Tidal effects determine the last orbits of inspiral E bind E 3PN bind = a fgw 1000Hz n, where n = The end of inspiral corresponds to marginally stable orbit (ISCO), which may be observed in gravitational waveforms. ISCO is given by an orbital instability for binary strange quark stars and by the mass-shedding limit for neutron star binaries. The frequency of GW at the end of inspiral phase strongly depends on EOS: khz for NS, khz for the MIT Bag model SS (Bejger et al. 2005, Limousin,Gondek-Rosińska, Gourgoulhon, 2005, Gondek-Rosinska & Limousin, 2008).

10 The f GW at the ISCO for equal mass BNS and BSS with M star = 1.35M 10 1,6 1,4 polytropic EOS, Γ=2 (Faber et al. 2002) strange stars (Gondek-Rosinska & Limousin 2008) neutron stars (Bejger et al. 2005) f GW,ISCO [khz] 1,2 1,0 0,8 0,6 0,12 0,14 0,16 0,18 0,2 M/R [Gondek-Rosińska & Limousin, 2009] We can determine the individual masses of the two neutron stars in the system taking into account the frequency evolution of the gravitational signal at the inspiral phase and high-order PN effects on the phase evolution of the signal (Cutler and Flanagan 1994). In addition we can get the compactness parameter M/R of neutron stars.

11 The impact of the total mass on the last orbits of inspiral M 1 = M ,1 0,08 SQS, the MIT bag model NS (Oechslin et al. 2004) NS with QM core (Oechslin et al. 2004) NS with M=1.35 M sol (Bejger et al. 2005) 0,08 SQSB56 SQSB60 SQSB40 Ω ISCO M 0,06 0,04 Ω ISCO M 0,06 0,04 0,02 0,02 0,10 0,15 0,20 M/R 0,10 0,15 0,20 0,25 M/R [Gondek-Rosinska & Limousin, 2009] The dependence of the frequency of gravitational waves at the ISCO on the compaction parameter for the equal-mass binaries can be described by the same simple analytical formulae (given by the power series of the compaction parameter) ΩM = A(M/R) B(M/R) 2.5,where A = 0.6 and B = 0.665, for broad ranges of masses independently on the strange star model.

12 The energy spectrum of GW emitted by BNS and BSS, M 1 = M 2 = 1.35M 12 E(f) de/df break frequencies: f 25, f 40, f 65 de bind /df GW [M sol Hz -1 ] 2e-05 1e-05 5e-06 2e-06 [Bejger et al. 2005] 1e f GW [Hz] [Gondek-Rosinska & Limousin, 2009] We can get the M/R of neutron stars based on the observed deviation of the gravitational energy spectrum from point-mass behavior at the end of inspiral (Faber et al. 2002, Gondek-Rosinska et al. 2007) NS f 40 = 0.79 khz (M/R = 0.14), 1.0 khz (0.176), 1.2(0.19) SS f 40 = 0.87 khz (M/R = 0.16), 1.02 khz (0.18), 1.1(0.19)

13 Measuring the EOS stiffness 13

14 C1. Comparison with polytropic EOS 14 NS EOS: Akmal et al (M/R = 0.176), SQSB60 (M/R = 0.187) Akmal et al EOS polytrope, γ=2 polytrope, γ= Etoiles de quarks étranges Etoiles à neutrons : polytrope 3PN non resum. irrot. (Blanchet 2002) (M ADM -M 0 ) [M sol ] f GW [Hz] [Bejger et al. 2005] M ADM - M 0 [M sol ] f GW [Hz] [Gondek-Rosinska & Limousin, 2009]

15 C2.Beyond the IWM approximation 15

16 C3. Radio vs Gravitational waves 16 Selection effects are widely different Can the observed populations be different? GW observability - Chirp mass - Coalescence times - the last seconds of their life Radio observability: - Radio Luminosity L > L min - Radio Loud Lifetime (t R,1, t R,2 ) - Period of BNS > 2h (Faulkner, 2004)

17 C3. Expected masses of merging neutron star binaries observed in gravitational waves (population synthesis) 17 Mmax=2.0 Mmax=2.5 Mmax=3.0 M M M q q q

18 Conclusions 18 Binary neutron stars observed in radio are a fraction of all binary NS. The expected masses of BNS in radio and GW may be different (0.7 < q < 1). GW observations of the last orbits of inspiral of BNS could yield important information about the equation of state of dense matter. The frequency of ISCO strongly depends on EOS ( khz for NS, khz for the MIT Bag model SS and > 2.4 khz for the Dey et al model - for equal mass of 1.35 Msol systems. The frequency of ISCO increases with increasing mass of a binary system. ISCO is given by an orbital instability for binary strange quark stars and by the mass-shedding limit for neutron star binaries. The late inspiral is dominated by high order tidal effects. Can we determine break frequencies from the energy spectrum? The crust plays important role

19 Faber et al

20 Radio vs Gravitational waves 20 Selection effects are widely different Can the observed populations be different? GW observability - Chirp mass - Coalescence times - the last seconds of their life Radio observability: - Radio Luminosity L > L min - Radio Loud Lifetime (t R,1, t R,2 ) - Period of BNS > 2h (Faulkner, 2004)

21 The relativistic binary merging plane 21 [Lorimer,D.R., the radio selected sample includes the long lived pulsars while a fraction of the merging binary neutron star population do not satisfy these conditions (Belczynski & Kalogera 2001, Bulik, Gondek-Rosinska, Belczynski, 2004, Gondek-Rosinska et al. 2005). It is necessary to perform the population synthesis calculations to find the most realistic masses of NS in binary systems let s model NS-NS and compare with radio populations

22 II. Searching for the most realistic masses of coalescing NS - The expected masses of BNS observed in gravitational waves 22 The results of the population synthesis calculations obtained with one of models of stellar evolution (using StarTrack population synthesis code developed by Belczyński, Bulik and Kalogera 2002,2008).Strong dependence of the results on: 1. initial mass spectrum of NS, 2. accretion rate in CE The blue line corresponds to all binary neutron stars to be observed in GW while the red line shows the contribution of the population of binary radio pulsars (Gondek-Rosińska, Bulik, Belczyński, 2005). In this model the population of neutron star binaries observed in gravitational waves are dominated by the systems with small mass ratio q 0.7, containg a star with canonical mass and a star with the mass close to the maximal mass. It is necessary to verify the results using diffent models and taking into account selection effects in observing radio pulsars (Os lowski,gondek-rosinska,bulik, Belczyński, 2009).

23 Expected masses of merging neutron star binaries observed in gravitational waves 23 Mmax=2.0 Mmax=2.5 Mmax=3.0 M M M q q q

24 Conclusions 24 Binary neutron stars observed in radio are a fraction of all binary NS. The expected masses of BNS in radio and GW may be different. It is necessary to calculate the expected masses of BNS observed in GW using different models and taking into account selection effects in observing radio pulsars. The frequency of ISCO strongly depends on EOS ( khz for NS, khz for the MIT Bag model SS and > 2.4 khz for the Dey et al model - for equal mass of 1.35 Msol systems. The frequency of ISCO increases with increasing mass of a binary system. ISCO is given by an orbital instability for binary strange quark stars and by the mass-shedding limit for neutron star binaries. GW observations of the last orbits of inspiral of BNS could yield important information about the equation of state of dense matter. The merger process of binary neutron stars and the gravitational waveforms depend strongly on the stellar mass ratio (Shibata, Taniguchi and Uryu 2003). We plan to extended current equal mass binary NS calculations to include the most realistic masses of components

25 Preparing a good input for GW detectors 25 What can we get from binary NS gravitational wave signal? What are expected masses of binary neutron stars observed in GW? Observations Population synthesis method - a numerical tool: StarTrack code [Belczyński, Kalogera, Bulik, 2002, 2008] Up to now all relativistic calculations of last orbits of binary neutron stars (BNS) were performed for mass ratio close to one (e.g. K. Taniguchi and E. Gourgoulhon 2003, Uryu and Eriguchi 2000, Gondek-Rosinska et al. 2007, Bejger et al. 2005, Oechslin et al. 2004) What is the impact of EOS on the last orbits of BNS? (the GW waveform are distinguishable from PN if d < 100 Mpc [J.Read et al. 2009, arxiv: ] quasi-equilibrium sequences, realistic EOS, irrotational flow A numerical tool: C++ library LORENE [ Up to now, all relativistic calculations (except Oechslin et al. 2004) of the hydrodynamical inspiral phase have been done for the simplified equation of state of dense matter, for the polytropic EOS. What is the effect of diferential rotation and EOS on the maximum mass of NS? Equilibrium models in GR, the rotation law (Komatsu et al.1989) A numerical tool [Ansorg, Gondek-Rosińska, Villain, arxiv: ]

26 I. Searching for the most realistic masses of coalescing NS - The radio observations B B J J B C J J J J The observed sample exhibits a strong peak for the mass ratio close to unity ( M NS 1.35M ),and a possible long tail stretching down to smaller values q 0.7.

27 Population Synthesis: The Method 27 Population synthesis is a Monte Carlo method which allows to evolve large stellar ensembles (both single and binary stars): start with initial conditions follow single and binary evolution calibrate results extract your population In the end synthetic populations are generated: statistical analysis, comparisons with observations, modifications to the input physics Research groups: Belczynski, Bulik, Kalogera... (StarTrack PS code) Lipunov, Postnov, Prokhorov Nelemans, van den Heuvel, Yungelson... Podsiadlowski..

28 Population synthesis method 28 Description of single star evolution (starts at ZAMS ends at formation of a compact object: WD (He, C-O, O-Ne), NS, BH based on formulae of Hurley et al M, R, L depend on M i and t M i [M ] < 8 WD; otherwise NS or BH Description of binary evolution - wind mass loss, mass transfer, common envelope evolution, detailed supernova explosion treatment, magnetic breaking, tidal circulation of binary orbit, GW energy loss in compact object binaries... uncertainties several models considered to verify robustness of the results Distribution of Initial Binary Parameters -M i, mass ratio q, separation a, eccentricity e e.g. Prob(M) M 2.5 ; Prob(q) const; Prob(e) e; Prob(a) 1 a

29 Nuclear EOS and the last orbits of binary neutron stars with masses 1.35M Akmal et al EOS BPAL12 EOS GlendNH3 EOS 3PN non resum. irrot. (Blanchet 2002) (M ADM - M 0 ) [M sol ] Energy emitted in gravitational waves vs frequency of GW (twice the orbital frequency) for binary neutron stars described by three different EOS of nuclear matter. The end of each sequence - a mass transfer between stars f GW [Hz] The end of quasiequlibrium sequence strongly depends on the compaction parameter M/R of NS (0.14; 0.176; 0.19)

30 polytrope, γ=2.5 polytrope, γ=2 Akmal et al EOS M=1.35, M/R= M [M sol ] R [km] Mass-radius relation of static models in isolation

31 AkmalPR EOS and polytropic EOS: evolutionary sequences Akmal et al EOS polytrope, γ=2 polytrope, γ=2.5 (M ADM -M 0 ) [M sol ] f GW [Hz] Gravitational mass at infinite separation : M 0 /2 = 1.35 M (M/R = 0.176)

32 Strange quark stars: evolutionary sequences E bin PN (irrot. Blanchet 2002) > SS POLYTROP f GW [Hz] Gravitational mass at infinite separation : M 0 /2 = 1.35 M (M/R = 0.187) [Gondek-Rosińska, Gourgoulhon, Limousin, in preparation]

33 The energy spectrum of GW emitted by BNS and BSS 33 E(f) de/df break frequencies: f 25, f 40, f 65 de bind /df GW [M sol Hz -1 ] 2e-05 1e-05 5e-06 2e-06 [Bejger et al. 2005] 1e f GW [Hz] [Gondek-Rosinska & Limousin, 2008] We can determine the individual masses of the two neutron stars in the system taking into account the frequency evolution of the gravitational signal at the inspiral phase and high-order PN effects on the phase evolution of the signal (Cutler and Flanagan 1994). In addition we can get the compactness parameter M/R of neutron stars based on the observed deviation of the gravitational energy spectrum from point-mass behavior at the end of inspiral (Faber et al. 2002, Bejger et al. 2005, Gondek-Rosinska et al. 2007, Gondek-Rosinska & Limousin, 2009).