Effects of U boson on the inner edge of neutron star crusts

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1 Effects of U boson on the inner edge of neutron star crusts Reporter : Hao Zheng ( 郑皓 ) ( INPAC & Department of Physics, Shanghai Jiao Tong University) Supervisor : Lie-Wen Chen

2 Outline Neutron Star introduction The possible non-newtonian gravity in Neutron Stars Methods used for calculation Results Conclusion Hao Zheng & Lie-Wen Chen, Phys. Rev. D 85, (2012) D. H. Wen, B. A. Li, and L. W. Chen, Phys. Rev. Lett. 103, (2009)

3 Neutron Stars Introduction The possible non-newtonian gravity in Neutron Stars Methods used for calculation Results Conclusion

4 Neutron Stars Introduction to the global properties of Neutron Stars & infinite nuclear matters J. M. Lattimer & M. Prakash, Science 304, 536 (2004)

5 Introduction to the global properties of Neutron Stars & infinite nuclear matters The mass-radius relation for static neutron stars

6 Introduction to the global properties of Neutron Stars & infinite nuclear matters The crustal fraction of moment of inertia for neutron stars (ΔI/I 0.014) B.Link, R.I.Epstein, and J.M.Lattimer, Phys.Rev.Lett.83, 3362 (1999)

7 Neutron Stars Introduction The possible non-newtonian gravity in Neutron Stars Methods used for calculation Results Conclusion

8 Introduction to the possible non-newtonian gravity in Neutron Stars The new holder of neutron star maximum mass from PSR J P. Demorest, T. Pennucci, S. Ransom, M. Roberts, & J. Hessels, Nature, 467, 1081 (2010)

9 Non-Newtonian gravity Introduction to the possible non-newtonian gravity in Neutron Stars E.G. Adelberger, B.R. Heckel and A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003).

10 Introduction to the possible non-newtonian gravity in Neutron Stars The Yukawa form due to the exchange of light vector U boson P.Fayet, Phys. Lett (1980); Nucl. Phys (1981) The possible existence of a neutral weakly coupled light spin-1 gauge U boson, which can be mediator of the putative fifth force providing a providing a possible mechanism for non-newtonian gravity: V UB () r = 2 g e 4π r μr Y. Fujii, Nature (London) (1971) The U boson can provide annihilation of light dark matter that can be responsible for the excess flux of 511keV photons coming from the central region of our Galaxy observed by the SPI/ENTEGRAL satellite. P. Jean et al., Astron. Astrophys (2003) C. Boehm, D. Hooper, J. Silk, etc, PRL (2004) C. Boehm, P. Fayet, and J. Silk, PRD (2004) C. Boehm and P. Fayet, Nucl. Phys. (2004)

11 Introduction to the possible non-newtonian gravity in Neutron Stars Various upper limits on the deviation from the inverse-square law have been put forward down to fm range. Jun Xu, Bao-An Li, Lie-Wen Chen & Hao Zheng, arxiv:

12 Neutron Stars introduction The possible non-newtonian gravity in Neutron Stars Methods used for calculation Results Conclusion

13 Methods used for calculation How to link nucleon-nucleon interaction(vn-n) to the global properties of stable nuclei (EOS, ρt, massradius relation & moment of inertia for NSs)?

14 Methods used for calculation Skyrme-Hartree-Fock Approach Skyrme s parametrized effective force r r r v(, ) = t (1 + x P ) δ ( ) central term σ 1 r 2 r r2 r + t1 (1 + x1p ) σ P δ ( ) + P δ ( ) 2 r r r + t2(1 + x2p ) P σ δ ( ) P non local term r r r r + iw0σ P δ( ) P spin orbit term 1 r σ r + t3 (1 + x3pσ ) ρ ( R ) δ ( ) density dependent term 6 r 2 μ /( hc) g e + r hc non Newtonian gravity 4π v = t δ( r r ) δ( r r ) (3) D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626 (1972).

15 Methods used for calculation i = i i i r σ τ

16 Methods used for calculation For the inner product of the spatial state, by defining: r r φi( ) = ir r r r ρ = φ φ * ( r) i ( r) i( r) i being the space wave function for particle i and density respectly, we get: rr rr ij ij = ij rr rr ij drdr 3 3 r r r r r r r r 1 2 i, j i, j = r r ρ( ) ρ( ) drdr

17 Methods used for calculation The space exchange operator: The inner product is : P i j r r r r r with the off-diagonal density : = j i rr rr i j j i = i j rr rr j i d rd r 3 3 r r r r r r r r 1 2 i, j i, j = r r r r ρ(, ) ρ(, ) d rd r r r r r ρ(, ) = φ ( ) φ ( ) * 1 2 i i 1 i 2

18 Methods used for calculation i i = i j στ ( ) στ ( ) στ ( ) στ ( ) i j σ( τ) σ( τ) σ( τ) σ( τ) σ( τ) σ( τ) σ( τ), σ( τ) = δ 1 2 σ( τ), σ( τ) 1 2

19 Methods used for calculation We get : with the energy density : where E = Η () 3 r d r H = K + H + H + H + H 2 K = h, 2m τ δ ρ eff Hδ = t 0 (2 + x0) ρ (2x0 + 1)( ρp + ρn ), 4 1 σ Hρ = t3ρ (2 + x3) ρ (2x3 + 1)( ρp + ρn ), Heff = [ t1(2 + x1) + t2(2 + x2) ] τρ + [ t2(2x2 + 1) t1(2x1+ 1) ]( τ pρp + τnρn), H fin = [ 3 t1 (2 + x1 ) t2 (2 + x2 ) ] ( ρ ) [ 3 t1 (2 x1 1) t2 (2 x2 1) ] ( ρp ) ( ρn ), fin

20 Methods used for calculation How to deal with the finite range Yukawa interaction? For Skyrme: ρ ( rr r, r ) ρ ( r, r ) δ( r r ) drdr= ρ ( r ) ρ ( rdr r ) τ 1 2 τ τ τ For Yukawa: r r μ 1 2 r r e ρτ 2 1 r r 1 2 = 1 2 r r ρ ( rr, ) ( r, r) drdr??? τ

21 Methods used for calculation Density Matrix Expansion Approach Using the coordinate transformation r = ( r + r )/2, s = r r the off-diagonal density can be expanded as r r r r s r s ρ( 1, 2) = ρ +, 2 2 r r * r s r s = φi ( + ) φi( ) i 2 2 = r s ( ) r r φ ( ) φ ( ) * e r r i i i J. W. Negele and D. Vautherin, Phys. Rev. C 5, 1472 (1972)

22 Methods used for calculation Replacing the angular integral of the square of a density matrix by the integral of the square of the angle average of the integral matrix r r 1 r s r s ρ( ) = d sρ, 4π Ω r s sinh ( 1 2) 2 r r = r (, ) s ( 1 2) 2 Using the Bessel-function expansion ρ r r r 1 2 = = 1 2 with sinh( xy) 1 = + xy x 2 P ( ) 2n+ 1 iy Qn ( y ) = iy + 2 (4n 3) j2n+ 1( x) Qn( y ) n= 0

23 Methods used for calculation The integration of the off-diagonal density can be approximated by r 1 r μ 2 μ r r r r e 3 3 r r e s 3 3 τ( 1, 2) τ ( 2, 1) r r 1 2 τ( ) τ ( ) 1 2 s ρ rr ρ r r drdr ρ rρ r drds with r r ρτ( ) ρτ( ) r r ρ ( ) ρ ( k s) + 2 ρ ( ) ρ ( k s) g( k s) s τ SL τ τ SL τ τ 1 2 r r 3 2 r ρτ( ) ττ( ) + kτ ρτ( ) 4 5

24 Methods used for calculation The energy density for Yukawa term is D 3 Hgra c d r ρ r ρ r r r μ /( hc) W r r e = h ( ) ( ), 2 r r E 2 n 2 p 6 2 2/3 8/3 n 8/3 p Hgra = ( hc) πw [ ρn I2 + ρpi2 + (3 π ) ( ρn I1 + ρp I1 )] 5 + 2( hc) πw( I ρ τ + I ρ τ ) n p 1 n n 1 p p 1 I 1 I + ( hc) πw[( I + ρ )( ρ ) ( I + ρ )( ρ ) ]. 2 2 n p n 1 2 p n n 1 p p ρn ρp

25 Methods used for calculation Calculation for the transition density Transition density is the baryon number density that separates the liquid core from the inner crust in neutron stars. In principle, the transition density can be obtained from comparing relevant properties of the nonuniform solid crust and the uniform liquid core mainly consisting of β-equilibrium neutrons, protons, and electrons matter. In practice, a good approximation is to search for the density at which the uniform liquid first becomes unstable against small amplitude density fluctuations with clusterization. Here I briefly introduce 3 kinds of method.

26 Methods used for calculation The thermodynamical method The intrinsic stability condition P tot > υ μ 0, μ > qc υ 0. which is equivalent to the following condition for infinite uniform NM V ther p 2 p p p 2 2 ρ xp ρ xp E( ρ, x ) E ( ρ, x ) E( ρ, x ) E( ρ, x ) = 2ρ + ρ ρ > 0 ρ

27 Methods used for calculation The curvature matrix method The instability region of homogeneous nuclear matters against clusterization is determined by introducing a finite-size spatially periodic density fluctuation to the system and then examining how the system free energy varies with fluctuation. f f 1 f ρ = ρ + δρ + δρ δρ + L 0 ( q) f( q) q q q q n, p, e ρ 2 = q q, q = n, p, e ρ 0 q ρ q 0 For a stable homogeneous npe matter system, we require the positive definiteness of the following curvature matrix C f CM μn μn μ n ρn ρp ρe μ p μp μ p = ρn ρp ρe μe μe μe ρn ρp ρ e μ q f = q= n, p, e ρ q

28 Methods used for calculation The Vlasov equation method For a β-stable and electrically neutral npe matter, the Vlasov equation is r r r r dfq( i, pi, t) fq( i, pi, t) r r r r r = + υqi r fq ( i, pi,) t r Uqi r p fq ( i, pi,) t = 0. dt t After linearizing the Vlasov equation, we can reexpress the above equation as a function of the collective density fluctuation with f T C ( δρ, δρ, δρ ) = 0 VE n p e LX n n 0 0 Un / ρn Un / ρp Un / ρ e f CVE = 0 Lp X p 0 U p / ρn U p / ρp U p / ρe LX e e Ue / ρn Ue / ρp Ue / ρ e 0 0 1

29 Neutron Stars introduction The possible non-newtonian gravity in Neutron Stars Methods used for calculation Results Conclusion

30 Results The core-crust transition density and pressure Transition density and pressure in neutron stars as functions of the L parameter with the MSL interaction using the three kinds of methods mentioned in the previous section.

31 Results U boson effects the nuclear matter system mainly through the combination g 2 /μ 2. But the effects of the U boson calculated by the CM & Vlasov equation methods is not only on the ratio g 2 /μ 2 but also the U- boson mass.

32 In order to see the U-boson mass dependence of ρt and Pt at fixed value of g 2 /μ 2, we display the 1/μ dependence of ρt and Pt from the CM & Vlasov equation methods with MSL0 interaction for g 2 /μ 2 = 75 GeV -2. Results

33 The mass-radius relation and crustal fraction of moment of inertia The EOS for different parts of a neutron star U boson can significantly stiffen the nuclear matter EOS The difference of the EOS for different values of g 2 /μ 2 is essentially due to the variation of Pt. Results

34 Results Solving the Tolman- Oppenheimer-Volkoff equations, we obtain the mass-radius relations. The neutron star mass can be enhanced strongly if the U boson are considered. The neutron star maximum mass can reach 2.07 M with g 2 /μ 2 = 75 GeV -2

35 Results The crustal fraction of total moment of inertia of a static neutron star is a particularly interesting quantity as it can be inferred from observations of pulsar glitches rotating neutron stars. The lower limit ΔI/I = has been constraint for the Vela pulsar.

36 Neutron Stars introduction The possible non-newtonian gravity in Neutron Stars Methods used for calculation Results Conclusion

37 Conclusion We have investigated effects of the light vector gauge U boson, that is weakly coupled to nucleons, on the core-crust transition density ρt and pressure Pt of neutron stars. Our results have shown that the ρt and pressure Pt depend on not only the ratio of coupling strength to mass squared of the U boson g 2 /μ 2 but also its mass μ. Both g 2 /μ 2 and μ can have significant influence on the massradius relation and the crustal fraction of total moment of inertia of neutron stars. Astrophysical observations on neutron star structures can be potentially useful to constrain properties of the U boson, e.g., its mass μ and the coupling constant g to nucleons.

38 Thank You!