Constraints on braneworld from compact stars

Transcription

1 Constraints on braneworld from compact stars Daryel Manreza Paret, ICN-UNAM Aurora Pérez Martinez, ICIMAF, Cuba Ricardo. González Felipe, ISEL, Portugal R. Gonzales Felipe, D. Manreza Paret and A. Perez Martinez, Eur. Phys. J. C (6) 76:7 (arxiv:6.97) ( ICN-UNAM) February, 7 / 9

2 Introduction. Compact Stars. White Dwarfs ( ICN-UNAM) Neutron Stars/Strange Stars February, 7 / 9

3 Introduction. Compact Stars. Third type of Compact Object: Black Holes ( ICN-UNAM) February, 7 / 9

4 s Introduction. Compact Stars. hyperon star strange star quark hybrid star s u p e r c o color superconducting strange quark matter (u,d,s quarks) SC CSL gcfl LOFF Σ,Λ,Ξ, n,p,e, µ n d u,d,s quarks SC CFL H CFL CFL K + CFL K CFL π N+e N+e+n u c t n,p,e, µ K i n π g p r o t o n traditional neutron star n superfluid R ~ km crust neutron star with pion condensate Fe 6 g/cm g/cm 4 g/cm Hydrogen/He atmosphere nucleon star F. Weber. Progress in Particle and Nuclear Physics, 54:9-88 (5). ( ICN-UNAM) February, 7 4 / 9

5 Introduction. Compact Stars. Quark Stars. Bodmer-Witten-Terazawa conjeture E / A (MeV) ud quark matter Fe 56 Nuclear matter (neutrons and protons) uds quark matter (strange matter) ~ ε / ε Picture from: F. Weber. Progress in Particle and Nuclear Physics, 54:9-88 (5). A. R. Bodmer. Phys. Rev. D,(97). E. Witten. Phys. Rev. D (984). H. Terazawa. Journal of the Physical Society of Japan, (989) ( ICN-UNAM) February, 7 5 / 9

6 Introduction. Compact Stars. Neutron Stars: Natural Laboratories Renxin Xu. J. Phys. G: Nucl. Part. Phys. 6 (9) 64 (9pp). ( ICN-UNAM) February, 7 6 / 9

7 Introduction. Compact Stars. Accedido ( ICN-UNAM) February, 7 7 / 9

8 Tolman-Oppenheimer-Volkoff equations The static, structure equations for a spherical symmetric relativistic star are found by solving Einsteint s equation G µν = R µν Rg µν = κ T µν. () For a spherically symmetric star, the metric is given by ds = e Φ(r) dt + e Λ(r) dr + r (dθ + sin θ dφ ). () and the energy momentum tensor Tolman Openheimer Volkof (TOV): T µν = ρu µ u ν + p (g µν + u µ u ν ) () dm dr dp dr = 4πr ρ, = G (ρ + p)(m + 4πpr ) r, Grm with initial conditions m() =, p() = p c and at stellar surface p(r) =. ( ICN-UNAM) February, 7 8 / 9

9 Tolman-Oppenheimer-Volkoff equations Equations of state P (MeV/fm ) 4 P=ε Stiffest possible equation of state Schroedinger based (p,n) RBHF (p,n) RH (p,n,h) RH (p,n,h,k ) RH (p,n,h,q) RH (p,n,h,dibaryons) Relativistic neutron gas (too soft) ε (MeV/fm ) F. Weber. Progress in Particle and Nuclear Physics, 54:9-88 (5). ( ICN-UNAM) February, 7 9 / 9

10 Tolman-Oppenheimer-Volkoff equations Mass-Radius diagram P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels. Nature, 467:88 (). ( ICN-UNAM) February, 7 / 9

11 Tolman-Oppenheimer-Volkoff equations Theoretical constrains GR: R > GM c. GM c P < : R > 9 4 Causality: A sound signal cannot propagate faster than the speed of light v < dp/dρ c R >.9 GM c Rotation: R > R max excluded by the 76 Hz pulsar J ad from the empirical result R <.4( J. M. Lattimer, M. Prakash, Science 4, 56 (4) Hz ) / ( M ) / km. ν M ( ICN-UNAM) February, 7 / 9

12 Tolman-Oppenheimer-Volkoff equations Observational constraints For neutron star radii 7.6 km R.9 km, (4) For neutron star masses.8m M M, (5) S. Guillot, M. Servillat, N.A. Webb, R.E. Rutledge, Astrophys. J. 77, 7 () K. Hebeler, J.M. Lattimer, C.J. Pethick, A. Schwenk, Phys. Rev. Lett. 5, 6 (). F. Ozel, D. Psaltis, R. Narayan, A.S. Villarreal, Astrophys. J. 757, 55 (). J. Antoniadis et al., Science 4, 6 (). ( ICN-UNAM) February, 7 / 9

13 Brane World Models From a classical point of view brane world models can be realised via the localization of matter and radiation fields on the brane, with gravity propagating in the bulk. Image from: Cavaglia, M., Int. J. Mod.Phys. A, 8, 84-88, (). [hep-ph/96]. ( ICN-UNAM) February, 7 / 9

14 Brane World Models Randall Sundrum Brane-Worlds The bulk is a portion of a 5-D anti-de Sitter (AdS 5 ) spacetime (extra dimension is curved rather than flat). What prevents gravity from leaking into the extra dimension at low energies is a negative bulk cosmological constant Λ 5 = 6/l where l is the curvature radius. The brane gravitates with self-gravity in the form of a brane tension λ. Randall&Sundrum PRL 999; Maartens, PRD ; Shiromizu et al PRD. ( ICN-UNAM) February, 7 4 / 9

15 TOV equations on the brane where G µν = R µν Rg µν = κ T µν + 6κ λ S µν E µν, (6) T µν = ρu µ u ν + p (g µν + u µ u ν ) (7) and S µν = ρ u µ u ν + ρ(ρ + p)(g µν + u µ u ν ), (8) where u µ is the four-velocity of the fluid. The tensor E µν reduces to the form E µν = 6 κ λ [ Uu µ u ν + Pr µ r ν + ] (U P) (g µν + u µ u ν ), (9) U y P are dark energy and pressure respectively. S µν local correction term. E µν non-local correction term. When λ, we recover GR. ( ICN-UNAM) February, 7 5 / 9

16 TOV equations on the brane where and, dm dr dp dr dφ dr du dr = 4πr ρ eff, () = (ρ + p) dφ dr, () = Gm + κ r [ p eff + (4P)/(κ 4 λ) ], () r(r Gm) = κ4 (ρ + p) dρ dr dp dr 6 dφ P (P + 4U) r dr, () ρ eff = ρ loc + 6 κ 4 λ U, p eff = p loc + κ 4 λ U. (4) ρ loc = ρ + ρ λ, p loc = p + pρ λ + ρ λ, (5) We need p(ρ) y P(U) and initial conditions: m() =, y p() = p c. At stellar surface p(r) = m(r) = M. For the dark component U, we shall assume U() =. ( ICN-UNAM) February, 7 6 / 9

17 TOV equations on the brane Equations of state In our analysis, the non-local dark components are modelled via the simple linear proportionality relation P = w U between the dark energy U and dark pressure P. For dense nuclear matter, we shall consider the analytical representation for the unified Brussels-Montreal EoS models, which are based on the nuclear energy-density functional theory with generalized Skyrme effective forces. For quark matter, we shall employ the simple phenomenological parametrisation which includes QCD and strange-quarkmass corrections. Hybrid EoS to study hybrid stars, i.e., stars with a hadronic outer region surrounding a quark (or mixed hadron-quark) inner core. A.Y. Potekhin, A.F. Fantina, N. Chamel, J.M. Pearson, S. Goriely, Astron. Astrophys. 56, A48 (). M. Alford,M. Braby,M.W. Paris, S. Reddy, Astrophys. J. 69, 969 (5). ( ICN-UNAM) February, 7 7 / 9

18 TOV equations on the brane Results: Neutron Stars Requiring agreement with observational constraints leads to a lower bound on the brane tension, λ 8 MeV/fm. The star radii lie in the range 8 km. M/M observational range PSR J48+4 NS BSk (GR) λ =5, w = λ = λ =5 λ = 4 λ =5 4 M/M NS BSk (GR) λ =5, w = λ = λ =5 λ = 4 λ = R [km] ρc/ρ ( ICN-UNAM) February, 7 8 / 9

19 TOV equations on the brane Results: Neutron Stars Mass-radius curves bend clockwise for w =.6 and w =.5. We have indicated with crosses ( ) the mass-radius configuration at which the GR causality condition v s is violated in such cases. M/M observational range PSR J48+4 NS BSk (GR) λ =6, w = w = w = w =.6 w =.5 M/M.5.5 NS BSk (GR) λ =6, w = w = w = w =.6 w = R [km] ρc/ρ ( ICN-UNAM) February, 7 9 / 9

20 TOV equations on the brane Results: Neutron Stars The maximum star mass predicted for this type of EoS is compatible with observations provided that λ 6 MeV/fm. For w., the value of M max remains practically constant with the variation of w, depending only on the value of λ. For. < w <., the maximum mass is quite sensitive to w. Mmax/M Mmax/M.5 NS BSk, w = w =5 w = w =.7 w = λ [MeV/fm ].5 NS BSk, λ =5 λ = λ =5 λ = 4 λ = w ( ICN-UNAM) February, 7 / 9

21 TOV equations on the brane Results: Quark Stars Agreement with observational constraints imposes the lower bound λ 4 MeV/fm, for w = The star radii lie in the range 8 km. M/M observational range PSR J48+4 QS, B =6(GR) λ =5, w = λ = λ =5 λ = 4 λ =5 4 M/M QS, B =6(GR) λ =5, w = λ = λ =5 λ = 4 λ = R [km] ρc/ρ ( ICN-UNAM) February, 7 / 9

22 TOV equations on the brane Results: Quark Stars For certain negative values of w the mass-radius curves bend clockwise, reaching the maximum mass at relatively high central densities, ρ c 4ρ, bounded by the requirement of subluminality of the EoS. M/M observational range PSR J48+4 QS, B =6(GR) λ =6, w = w = w = w =.6 w =.5 M/M QS, B =6(GR) λ =6, w = w = w = w =.6 w = R [km] ρc/ρ ( ICN-UNAM) February, 7 / 9

23 TOV equations on the brane Results: Quark Stars The maximum star mass predicted for this type of EoS is compatible with observations provided that λ MeV/fm. For w., the value of M max remains practically constant with the variation of w, depending only on the value of λ. For. < w <., the maximum mass is quite sensitive to w. QS, B = 6, λ =5 λ = λ =5 λ = 4 λ =5 4 Mmax/M Mmax/M.5 QS, B = 6, w = w =5 w = w =.7 w = λ [MeV/fm ] w ( ICN-UNAM) February, 7 / 9

24 TOV equations on the brane Results: Hybrid Stars Requiring agreement with observational constraints leads to a lower bound on the brane tension, λ 8 MeV/fm. The maximum mass M.98M is obtained for GR, and this value is consistent with the observational range of the pulsar PSR J48+4. M/M observational range PSR J48+4 HS (GR) λ =5, w = λ = λ =5 λ = 4 λ =5 4 M/M HS (GR) λ =5, w = λ = λ =5 λ = 4 λ = R [km] ρc/ρ ( ICN-UNAM) February, 7 4 / 9

25 TOV equations on the brane Results: Hybrid Stars As in the case of quark stars,we notice that the clockwise bending of the mass-radius curves persists for certain negative values of w, reaching the maximum mass at relatively high central densities,ρ c 45ρ M/M observational range PSR J48+4 HS (GR) λ =6, w = w = w = w =.6 w =.5 M/M R [km].5 HS (GR) λ =6, w = w = w = w =.6 w = ρc/ρ ( ICN-UNAM) February, 7 5 / 9

26 TOV equations on the brane Results: Hybrid Stars The maximum star mass predicted for this type of EoS is compatible with observations provided that λ MeV/fm. For w., the value of M max remains practically constant with the variation of w, depending only on the value of λ. For. < w <., the maximum mass is quite sensitive to w. HS, λ =5 λ = λ =5 λ = 4 λ =5 4 Mmax/M Mmax/M.5 HS, w = w =5 w = w =.7 w = λ [MeV/fm ] w ( ICN-UNAM) February, 7 6 / 9

27 Conclusions Compact Stars are natural laboratories to test new theories. In all the three EOS cases, the maximum mass and the corresponding star radius decrease as λ decreases. Furthermore, the central energy density ρ c required to achieve the maximum mass configuration is always less than that of GR ρ c 7ρ for pure neutron stars and quark stars slightly lower for hybrid stars, ρ c 4.5ρ The star radii lie in the ranges 8 km for pure neutron stars, 8 km for quark stars 9 4 km for hybrid stars. The maximum star mass as a function of λ and w was also studied for the three families of stars. Requiring agreement with observational constraints leads to a lower bound on the brane tension, λ MeV/fm for all three types of stars. ( ICN-UNAM) February, 7 7 / 9

28 Muchas Gracias ( ICN-UNAM) February, 7 8 / 9

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