FACULTY OF MATHEMATICAL, PHYSICAL AND NATURAL SCIENCES. AUTHOR Francesco Torsello SUPERVISOR Prof. Valeria Ferrari

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1 FACULTY OF MATHEMATICAL, PHYSICAL AND NATURAL SCIENCES AUTHOR Francesco SUPERVISOR Prof. Valeria Ferrari

2 Internal structure of a neutron star M [ 1, 2] M n + p + e + µ 0.3km; atomic nuclei +e 0.5km; PRM +n + e H + K + π + q interaction (QCD):? gcm gcm gcm 3 0.5R 0.95R R = 9 15km

3 Internal structure of a neutron star M [ 1, 2] M n + p + e + µ 0.3km; atomic nuclei +e 0.5km; PRM +n + e H + K + π + q interaction (QCD):? C NS = GM NS c 2 R NS gcm gcm gcm 3 0.5R 0.95R R = 9 15km

4 Internal structure of a neutron star M [ 1, 2] M n + p + e + µ 0.3km; atomic nuclei +e 0.5km; PRM +n + e H + K + π + q interaction (QCD):? C NS = GM NS c 2 R NS 0.1 C = GM c 2 R 1.47km km gcm gcm gcm 3 0.5R 0.95R R = 9 15km

5 Internal structure of a neutron star M [ 1, 2] M n + p + e + µ 0.3km; atomic nuclei +e 0.5km; PRM +n + e H + K + π + q interaction (QCD):? gcm 3 0.5R 10 7 gcm gcm R R = 9 15km C NS = GM NS c 2 R NS 0.1 C = GM c 2 R 1.47km km 10 6 C NS 10 5 C C STRONG FIELD: GENERAL RELATIVITY IS NECESSARY

6 Binary systems NS-NS or NS-BH merging t GRAVITATIONAL WAVES EMISSION Initial phase: point-like approximation, the gravitational wave does not depend on the internal structure of neutron stars Final phase: neutron stars are deformed due to mutual tidal interaction and the gravitational wave depends on their internal structures

7 Gravitational waves revelation II generation AdvancedLIGO, AdvancedVIRGO III generation Einstein Telescope AdvancedLIGO Caltech, MIT AdvancedVIRGO Centre National de la Recherche Scientifique (CNRS), INFN Einstein Telescope EGO, INFN, Max-Planck-Institut für Gravitationsphysik, CNRS, University of Birmingham, University of Glasgow, NIKHEF, Cardiff University

8 Purpose of this thesis UNDESTANDING OF WHAT IS THE INTERNAL REGION OF A NEUTRON STAR THAT MAINLY DETERMINES ITS DEFORMABILITY, in order to establish what is the internal region of a neutron star about which we can obtain information by measuring gravitational waves emitted by binary systems shortly before the merging

9 Static and spherically symmetric neutron star with mass M and radius R (non rotating star) ds 2 = e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 ( dϑ 2 + sin 2 (ϑ)dϕ 2), with G = c = 1

10 Static and spherically symmetric neutron star with mass M and radius R (non rotating star) r R ds 2 = e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 ( dϑ 2 + sin 2 (ϑ)dϕ 2), with G = c = 1 TOV equations (Tolman-Oppenheimer-Volkoff) dm(r) = 4πr 2 ɛ(r) dr dp(r) dr = [ɛ(r) + p(r)] [ m(r) + 4πr 3 p(r) ] r (r 2m(r))

11 Static and spherically symmetric neutron star with mass M and radius R (non rotating star) r R ds 2 = e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 ( dϑ 2 + sin 2 (ϑ)dϕ 2), with G = c = 1 1 m(r) r 2 TOV equations (Tolman-Oppenheimer-Volkoff) dm(r) = 4πr 2 ɛ(r) dr dp(r) dr = [ɛ(r) + p(r)] [ m(r) + 4πr 3 p(r) ] r (r 2m(r)) 3 unknown functions e 2 ODE: (1 e 2λ(r)) 2 ɛ(r) = mass-energy density 3 p(r) = pressure

12 Static and spherically symmetric neutron star with mass M and radius R (non rotating star) r R ds 2 = e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 ( dϑ 2 + sin 2 (ϑ)dϕ 2), with G = c = 1 TOV equations (Tolman-Oppenheimer-Volkoff) dm(r) = 4πr 2 ɛ(r) dr dp(r) dr = [ɛ(r) + p(r)] [ m(r) + 4πr 3 p(r) ] r (r 2m(r)) 3 unknown functions e 2 ODE: 1 m(r) r (1 e 2λ(r)) THIRD EQUATION 2 2 ɛ(r) = mass-energy density = Equation of State 3 p(r) = pressure (EOS)

13 Static and spherically symmetric neutron star with mass M and radius R (non rotating star) r R ds 2 = e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 ( dϑ 2 + sin 2 (ϑ)dϕ 2), with G = c = 1 TOV equations (Tolman-Oppenheimer-Volkoff) dm(r) = 4πr 2 ɛ(r) dr dp(r) dr = [ɛ(r) + p(r)] [ m(r) + 4πr 3 p(r) ] r (r 2m(r)) 3 unknown functions e 2 ODE: 1 m(r) r (1 e 2λ(r)) THIRD EQUATION 2 2 ɛ(r) = mass-energy density = Equation of State 3 p(r) = pressure (EOS) r R e 2ν(r) = 1 2M r Schwarzschild metric e 2λ(r) = ( 1 2M r ) 1

14 Piecewise polytropics Read, Lackey, Owen, Friedman, Physical Review D79, , 2009 [ρ i 1, ρ i ], with i = 1,..., n and n number of intervals p(ρ) = K i ρ γ i : EOS is polytropic ( ) ( ) : the first principle ρ [ρ i 1, ρ i ] = ɛ 1 d = pd of Thermodynamics ρ ρ is valid p(ρ) γ 3 γ 2 p 2 p 1 γ 1 p c SLY ρ 0 centre ρ 2 = gcm 3 ρ gcm 3 ρ c. ρ = gcm 3 ρ surface

15 Tidal λ: definition TIDAL QUADRUPOLAR FIELD = l = 2 TIDAL QUADRUPOLAR TENSOR E ij = ( R tid µναβ 2 E m Yij 2m m= 2 e µ (i) eν (0) eα (j) eβ (0) ) = MASS QUADRUPOLE MOMENT Q ij = d 3 xδρ( x) (x i x j 13 ) r2 δ ij = 2 Q m Yij 2m m= 2

16 Tidal λ: definition TIDAL QUADRUPOLAR FIELD = l = 2 TIDAL QUADRUPOLAR TENSOR E ij = ( R tid µναβ 2 E m Yij 2m m= 2 e µ (i) eν (0) eα (j) eβ (0) ) = MASS QUADRUPOLE MOMENT Q ij = d 3 xδρ( x) (x i x j 13 ) r2 δ ij = 2 Q m Yij 2m m= 2 WEAK TIDAL FIELD: [ ] 2R 5 Q ij = 3G k 2 E ij = λ 2R5 3G k 2 FIRST PERTURBATIVE ORDER, O(E ij )

17 Tidal λ of a static and spherically symmetric neutron star in an external tidal quadrupolar gravitational field δg µ ν = 8πδT µ ν

18 Tidal λ of a static and spherically symmetric neutron star in an external tidal quadrupolar gravitational field δg µ ν = 8πδT µ ν Thorne, Physical Review D58, , 1998 g 00 = 1 + 2M ( + 3λ ) r r 3 r2 E 0 Y 20 (ϑ)

19 Tidal λ of a static and spherically symmetric neutron star in an external tidal quadrupolar gravitational field δg µ ν = 8πδT µ ν Thorne, Physical Review D58, , 1998 g 00 = 1 + 2M ( + 3λ ) r r 3 r2 E 0 Y 20 (ϑ) λ = ( 2R 5 ) 8C 5 3G 5 (1 2C)2 (2 + 2C(y 1) y) [2C (6 3y + 3C(5y 8)) + 4C 3 (13 11y + C(3y 2)+ +2C 2 (1 + y) ) + 3 (1 2C) 2 (2 y + 2C(y 1)) ln (1 2C)] 1, C = M R, y = RH (R) H(R) Hinderer, Astrophysical Journal 677: , 2008 H(r) is the radial perturbation of the Schwarzschild metric

20 Estimate of λ in the interior of the star APPROXIMATION We evaluate the tidal λ in r < R supposing that the internal region r < r R does not exist. To calculate λ( r) it suffices to substitute R r in the previous formula of λ λ λ( r) r R r R

21 Choice of EOS and of stellar configurations used in our study REASONS Extreme EOS: most and least compact neutron stars Different composition of matter in the interior of the stars: MS1: n, p, e, µ, relativistic mean field theory ALF2: n, p, e, µ, q, variational method ecc...

22 Choice of EOS and of stellar configurations used in our study REASONS Extreme EOS: most and least compact neutron stars Different composition of matter in the interior of the stars: MS1: n, p, e, µ, relativistic mean field theory ALF2: n, p, e, µ, q, variational method ecc... EOS in decreasing order of produced neutron stars compressibility χ = (γ i p) 1 : greater C = M R, R1, WFF1, APR4, Γ1, Γ2, ALF2, MS1 stiff, soft minor C = M R R1, Γ1, Γ2 are defined in Yagi, Stein, Pappas, Yunes, Apostolatos, Physical Review D90, , 2014

23 Radial profiles of tidal ( ) λ(r) ( r vs λ R) D = [0.5R, 0.98R]

24 Radial profiles of tidal ( ) λ(r) ( r vs λ R) D = [0.5R, 0.98R] What is the physical internal region (inner core, outer core, inner crust, outer crust) that mainly contributes to determine λ?

25 Tidal profiles depending on mass density ( ) λ(r) vs ρ(r) λ PHYSICAL REGION CONTRIBUTIONS TO λ (1.4M ) EOS stiff soft CORE 45% 80% CRUST 55% 20% INNER CORE OUTER CORE 0% 25% 45% 55%

26 Binary system NS-NS: gravitational wave h(f ) = A(f )e iψ(f ), with f wave frequency ψ(f ) = ψ pp (f ) +ψ Q (f ) +ψ λ (f ) point particle rotational tidal model s contribution: contribution: contribution Q = MQrot J 2 λ = λ M 5

27 Binary system NS-NS: gravitational wave h(f ) = A(f )e iψ(f ), with f wave frequency ψ(f ) = ψ pp (f ) +ψ Q (f ) +ψ λ (f ) point particle rotational tidal model s contribution: contribution: contribution Q = MQrot J 2 λ = λ M 5 ψ λ (f ) = (Mπf { ) 5/3 24 [( 1 + 7η 31η 2) λ S + + ( 1 + 9η + 11η 2) λ a δm ] x 5} + O(x 6 ) ( M = µ 3 (M 1 + M 2 ) 2) 1/5 M = chirp mass, µ = 1 M 2 M, η = 1 M 2 M 1 + M 2 (M 1 + M 2 ) 2,, x = ((M M M 2 )πf ) 5/3, λ M S = λ 1 + λ 2, λ a = λ 1 λ δm = M 1 M 2

28 Conclusions Future measure of λ by gravitational waves revelation

29 Conclusions Future measure of λ by gravitational waves revelation λ close to the value related to soft EOS

30 Conclusions Future measure of λ by gravitational waves revelation λ close to the value related to soft EOS λ is mainly determined by the star s core

31 Conclusions Future measure of λ by gravitational waves revelation λ close to the value related to soft EOS λ is mainly determined by the star s core EOS in the innermost region of the star is soft

32 Conclusions Future measure of λ by gravitational waves revelation λ close to the value related to soft EOS λ is mainly determined by the star s core EOS in the innermost region of the star is soft λ close to the value related to stiff EOS

33 Conclusions Future measure of λ by gravitational waves revelation λ close to the value related to soft EOS λ is mainly determined by the star s core EOS in the innermost region of the star is soft λ close to the value related to stiff EOS λ is mainly determined by the star s crust

34 Conclusions Future measure of λ by gravitational waves revelation λ close to the value related to soft EOS λ is mainly determined by the star s core EOS in the innermost region of the star is soft λ close to the value related to stiff EOS λ is mainly determined by the star s crust Little information about EOS in the innermost region