Neutron Stars as Laboratories for Gravity Physics

Transcription

1 Neutron Stars as Laboratories for Gravity Physics Cemsinan Deliduman Department of Physics, Mimar Sinan University, Turkey S. Arapoglu, C.D., K.Y. Ekşi, JCAP 1107 (2011) 020 [arxiv: ]. C.D., K.Y. Ekşi, V. Keleş, JCAP 1205 (2012) 036 [arxiv: ]. M.-K. Cheoun, C.D., C. Güngör, V. Keleş, C. Ryu, T. Kajino, G. Mathew, [arxiv: ].

2 Accelerated expansion of the universe Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

3 Accelerated expansion of the universe Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

4 Accelerated expansion of the universe Main assumptions of FRW Cosmology: 1 Universe is homogenous. 2 Universe is isotropic. 3 Einstein s theory of gravity (general relativity) is valid at all scales. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

5 Why General Relativity? GR passed from all solar system tests with great success. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

6 Why General Relativity? GR passed from all solar system tests with great success. GR fits the Hulse-Taylor pulsar data. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

7 Why General Relativity? GR passed from all solar system tests with great success. GR fits the Hulse-Taylor pulsar data. GR presents us the fundamental framework for understanding the expanding universe. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

8 Accelerated expansion of the universe Mannheim, [astro-ph/ ]: The q 0 = 0.37 conformal gravity fit (upper curve) and the Ω M (t 0 ) = 0.3, Ω Λ (t 0 ) = 0.7 standard model fit (lower curve) to the z < 1 supernovae Hubble plot data. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

9 Alternative gravity models and tests Contribution of dark energy Einstein tensor = Matter + Dark energy. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

10 Alternative gravity models and tests Contribution of dark energy Modification of gravity theory Einstein tensor = Matter + Dark energy. Einstein tensor + Modifications = Matter. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

11 Alternative gravity models and tests Contribution of dark energy Modification of gravity theory Einstein tensor = Matter + Dark energy. Einstein tensor + Modifications = Matter. Ways to modify the theory of gravity: 1 Modify the Einstein Hilbert action of general relativity w/o adding new gravitational degrees of freedom. 2 Add new fields that gravitate: Scalar(s), vector(s), spinor(s), new tensor(s), etc. 3 Change how matter fields and perhaps cosmological constant gravitates. 4 Change geometry, e.g. use Finsler, Wietzenböck, noncommutative geometry, etc. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

12 Alternative gravity models and tests Contribution of dark energy Modification of gravity theory Einstein tensor = Matter + Dark energy. Einstein tensor + Modifications = Matter. Ways to modify the theory of gravity: 1 Modify the Einstein Hilbert action of general relativity w/o adding new gravitational degrees of freedom. 2 Add new fields that gravitate: Scalar(s), vector(s), spinor(s), new tensor(s), etc. 3 Change how matter fields and perhaps cosmological constant gravitates. 4 Change geometry, e.g. use Finsler, Wietzenböck, noncommutative geometry, etc. Tests of gravitational theories: 1 In weak gravity regime: be compatible with the solar system tests and table top experiments. 2 In cosmological scales: produce the late time accelerated expansion, be free of gravitational instabilities, and obey constraints of the standard model of cosmology. 3 In strong gravity regime: have solutions for neutron stars with mass radius relation inside the current observational bounds. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

13 Strength of gravity The strongest field, SS tests can probe, corresponds to a compactness of ɛ GM R c , and to a spacetime curvature of ξ = GM R 3 c cm 2. A gravity test with neutron stars (M = 1.4M and R = 10 km) would probe a compactness ɛ GM Rc ɛ, and a spacetime curvature of ξ = GM R 3 c cm ξ. ξ=gm/r 3 c 2 (cm -2 ) NS XRB IMBH Sgr A * WD km AGN MS R R SS 1 AU M 10 6 M M 1M MW pc Dark Matter kpc Dark Energy (Psaltis, 2008) ε=gm/rc 2 Over the Horizon Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

14 Why neutron stars, and not black holes? The strongest gravitational fields around astrophysical systems can be found in the vicinity of neutron stars and black holes in X-ray binaries. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

15 Why neutron stars, and not black holes? The strongest gravitational fields around astrophysical systems can be found in the vicinity of neutron stars and black holes in X-ray binaries. BH solutions of modified theories of gravity are essentially indistinguishable from those of GR. (Psaltis et al., 2008) There are some exceptions, e.g. Chern-Simons theory. A phenomenon that occurs even just above the horizon of a BH cannot be used in testing GR against its alternatives, because all theories would make the same prediction for that phenomenon. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

16 Why neutron stars, and not black holes? The strongest gravitational fields around astrophysical systems can be found in the vicinity of neutron stars and black holes in X-ray binaries. BH solutions of modified theories of gravity are essentially indistinguishable from those of GR. (Psaltis et al., 2008) There are some exceptions, e.g. Chern-Simons theory. A phenomenon that occurs even just above the horizon of a BH cannot be used in testing GR against its alternatives, because all theories would make the same prediction for that phenomenon. Only NS are left for discriminating predictions of different theories. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

17 Constraints on theory of gravity from neutron stars In order to do the tests we should know EoS better than gravity. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

18 Constraints on theory of gravity from neutron stars In order to do the tests we should know EoS better than gravity. Do we? Uncertainties in EoS do not preclude significant constraints on the strong-field behavior of gravity. (DeDeo & Psaltis, 2003) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

19 Constraints on theory of gravity from neutron stars In order to do the tests we should know EoS better than gravity. Do we? Uncertainties in EoS do not preclude significant constraints on the strong-field behavior of gravity. (DeDeo & Psaltis, 2003) The density inside the neutron star is larger by only an order of magnitude compared to the densities probed by nuclear scattering data that are used to constrain the EoS. The curvature around a neutron star is larger by 15 orders of magnitude compared to the curvature probed by solar-system tests. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

20 Constraints on theory of gravity from neutron stars In order to do the tests we should know EoS better than gravity. Do we? Uncertainties in EoS do not preclude significant constraints on the strong-field behavior of gravity. (DeDeo & Psaltis, 2003) The density inside the neutron star is larger by only an order of magnitude compared to the densities probed by nuclear scattering data that are used to constrain the EoS. The curvature around a neutron star is larger by 15 orders of magnitude compared to the curvature probed by solar-system tests. Neutron stars can indeed be used in testing the strong-field behavior of a gravity theory! Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

21 Neutron Stars in Alternative Gravity Theories (DeDeo & Psaltis, 2003) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

22 A simple modification of general relativity f (R) gravity models S = 1 16π d 4 x gf (R) + S matter We set G = 1 and c = 1 until the numerical analysis. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

23 A simple modification of general relativity f (R) gravity models S = 1 16π d 4 x gf (R) + S matter We set G = 1 and c = 1 until the numerical analysis. Choice of f (R) f (R) = R + αh(r) + O(α 2 ) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

24 A simple modification of general relativity f (R) gravity models S = 1 16π d 4 x gf (R) + S matter We set G = 1 and c = 1 until the numerical analysis. Choice of f (R) f (R) = R + αh(r) + O(α 2 ) Field equations (1 + αh R )G µν 1 2 α(h h RR)g µν α( µ ν g µν )h R = 8πT µν, Here G µν = R µν 1 2 Rg µν is the Einstein tensor and h R = dh dr. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

25 Perturbative approach Shearless matter: Energy-momentum tensor T µν = Diag[ ρ(r), P(r), P(r), P(r)] Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

26 Perturbative approach Shearless matter: Energy-momentum tensor T µν = Diag[ ρ(r), P(r), P(r), P(r)] Spherical symmetry: Metric ds 2 = g µν dx µ dx ν = e 2φα dt 2 + e 2λα dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

27 Perturbative approach Shearless matter: Energy-momentum tensor T µν = Diag[ ρ(r), P(r), P(r), P(r)] Spherical symmetry: Metric ds 2 = g µν dx µ dx ν = e 2φα dt 2 + e 2λα dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) Perturbative expansions Metric : g µν = g (0) µν + αg (1) µν + O(α 2 ), φ α = φ + αφ 1 + and λ α = λ + αλ 1 +, Matter : ρ α = ρ + αρ 1 + and P α = P + αp 1 +. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

28 Perturbative forms of field equations Field equations take the following forms: 8πr 2 ρ α = 1 e 2λα (1 2rλ α) +αh R r 2 [8πρ ( h h R 8πr 2 P α = 1 + e 2λα (1 + 2rφ α) +αh R r 2 [8πP 1 2 ( h h R ) ( R e 2λ r 1 (2 rλ ) h R h R ) ] R + e 2λ r 1 (2 + rφ ) h R h R )] + h R h R Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

29 Perturbative forms of field equations Field equations take the following forms: 8πr 2 ρ α = 1 e 2λα (1 2rλ α) +αh R r 2 [8πρ ( h h R 8πr 2 P α = 1 + e 2λα (1 + 2rφ α) +αh R r 2 [8πP 1 2 ( h h R To match the solution outside the star: ) ( R e 2λ r 1 (2 rλ ) h R h R ) ] R + e 2λ r 1 (2 + rφ ) h R h R )] + h R h R Mass parameter e 2λα = 1 2M α r with M = 4π ρ(r)r 2 dr Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

30 Tolman Oppenheimer Volkoff equations Good job Robert. I am happy to see that the stars are in equilibrium without introducing any new constant. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

31 Tolman Oppenheimer Volkoff equations First modified TOV equation dm α dr = 4πr 2 ρ α 1 2 αh R [ 8πr 2 ρ + r 2 2 ( h h R R) +(4πρr 3 2r + 3M) h R h R r(r 2M) h R h R ] Conservation Equation µ T µν = 0 dp α dr = (ρ α + P α ) dφ α dr From the rr field equation (r 2M α ) dφα dr = 4πr 2 P α + Mα r 1 2 αh R [ 8πr 2 P + r 2 2 ( h h R R) +(2r 3M + 4πPr 3 ) h R h R ] Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

32 f (R) = R + αr 2 gravity model What could be the function f (R), or h(r) in f (R) = R + αh(r)? Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

33 f (R) = R + αr 2 gravity model What could be the function f (R), or h(r) in f (R) = R + αh(r)? For f RR (R) 0, in addition to the massless spin-2 graviton, f (R) gravity models contains a scalar particle. It is neither a tachyon nor a ghost for f RR (R) > 0 (Starobinsky, 1980). Graviton is not a ghost if f R (R) > 0. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

34 f (R) = R + αr 2 gravity model What could be the function f (R), or h(r) in f (R) = R + αh(r)? For f RR (R) 0, in addition to the massless spin-2 graviton, f (R) gravity models contains a scalar particle. It is neither a tachyon nor a ghost for f RR (R) > 0 (Starobinsky, 1980). Graviton is not a ghost if f R (R) > 0. Models with 1/R and R m terms seem to pass the Solar System tests, i.e. they have the acceptable Newtonian limit and no instabilities. Models with ln R terms do not seem to pass the Solar System tests (Nojiri & Odintsov, 2006). Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

35 f (R) = R + αr 2 gravity model What could be the function f (R), or h(r) in f (R) = R + αh(r)? For f RR (R) 0, in addition to the massless spin-2 graviton, f (R) gravity models contains a scalar particle. It is neither a tachyon nor a ghost for f RR (R) > 0 (Starobinsky, 1980). Graviton is not a ghost if f R (R) > 0. Models with 1/R and R m terms seem to pass the Solar System tests, i.e. they have the acceptable Newtonian limit and no instabilities. Models with ln R terms do not seem to pass the Solar System tests (Nojiri & Odintsov, 2006). f (R) = R + αr n models generically lead to the wrong expansion law: the usual matter era preceding the late-time accelerated stage does not have the usual a t 2/3 behavior but rather a t 1/2 which would obviously make these models cosmologically unacceptable (Amendola, Polarski & Tsujikawa, 2006). Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

36 f (R) = R + αr 2 gravity model What could be the function f (R), or h(r) in f (R) = R + αh(r)? For f RR (R) 0, in addition to the massless spin-2 graviton, f (R) gravity models contains a scalar particle. It is neither a tachyon nor a ghost for f RR (R) > 0 (Starobinsky, 1980). Graviton is not a ghost if f R (R) > 0. Models with 1/R and R m terms seem to pass the Solar System tests, i.e. they have the acceptable Newtonian limit and no instabilities. Models with ln R terms do not seem to pass the Solar System tests (Nojiri & Odintsov, 2006). f (R) = R + αr n models generically lead to the wrong expansion law: the usual matter era preceding the late-time accelerated stage does not have the usual a t 2/3 behavior but rather a t 1/2 which would obviously make these models cosmologically unacceptable (Amendola, Polarski & Tsujikawa, 2006). So we analyze the case of simplest R m model with m = 2: S = d 4 x g [ R + αr 2] + S matter Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

37 Numerical analysis Equations to be solved: dm dr = 4πr 2 ρ αr 2 K, ( dp GMρ 1 + P dr = r ( ) 2 1 2GM c 2 r ρc 2 ) ( 1 + 4πr 3 P + 1 ) Mc 2 2 αr 2 H K and H are complicated and given in terms of general relativistic quantities. A characteristic value of R: R 0 GM /c 2 R 3 Gρ c/c 2. As α 0 1/R 0 we infer that α 0 c 2 /Gρ c. For validity of perturbative approach: α/α 0 << 1. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

38 Numerical analysis Equations to be solved: dm dr = 4πr 2 ρ αr 2 K, ( dp GMρ 1 + P dr = r ( ) 2 1 2GM c 2 r ρc 2 ) ( 1 + 4πr 3 P + 1 ) Mc 2 2 αr 2 H K and H are complicated and given in terms of general relativistic quantities. A characteristic value of R: R 0 GM /c 2 R 3 Gρ c/c 2. As α 0 1/R 0 we infer that α 0 c 2 /Gρ c. For validity of perturbative approach: α/α 0 << 1. We use an analytical representation of log ρ (log P) for all the EoSs obtained by fitting the tabulated EoS data. We employ a Runge-Kutta scheme from the center of the star for a certain value of central pressure, P c. We record the mass and radius for each central pressure. We then repeat this procedure for a range of α to see the effect of the perturbative term we added to the Lagrangian. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

39 Observational Constraints Measured mass of PSR J : M = 1.97 ± 0.04 M (Demorest et al., 2010) Mass (solar) GR P < causality AP3 ENG AP4 J SQM3 J FSU SQM1 J PAL6 GM3 GS1 Double NS Systems MPA1 MS1 PAL1 MS2 MS rotation Nucleons Nucleons+ExoticStrange Quark Matter Radius (km) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

40 Observational Constraints Measured mass and radius of three neutron stars: 4U , 4U , EXO (Özel, Baym & Güver, 2010) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

41 Observational Constraints Measured mass and radius of three neutron stars: 4U , 4U , EXO (Özel, Baym & Güver, 2010) For a critic of these constraints: (Steiner et al., 2010) M (M ) R (km) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

42 Equation of State 1: FPS M * / M O R < R S EoS = FPS α 9 =-2.0 α 9 =-1.0 α 9 = 0.0 α 9 = 1.0 α 9 = R * (km) The observational constraints of [Özel at al., 2010] is shown with the thin black contour; the measured mass M = 1.97 ± 0.04 M of PSR J [Demorest et al., 2010] is shown as the horizontal black line with grey error bar. Each solid line corresponds to a stable configuration for a specific value of α. Dashed lines show the solutions for unstable configurations (dm/dρ c < 0). The grey shaded region shows where the total mass would be enclosed within its Schwarzschild radius. S. Arapoglu, C.D., K.Y. Ekşi, JCAP 1107 (2011) 020. (α 9 = α 10 9 ) Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

43 Equation of State 2: GS1 M * / M O R < R S EoS = GS1 α 9 =-4.0 α 9 =-2.0 α 9 = 0.0 α 9 = 1.0 α 9 = R * (km) For GS1 [Glendenning & Schaffner-Bielich, 1999], the maximum mass in GR remains well below the measured mass of PSR J The maximum mass of neutron stars for this EoS can reach up to 2 M for α 9 = 4. Starting from α 9 = 2 the stability condition (dm/dρ c > 0) is satisfied for the whole range of central densities considered. S. Arapoglu, C.D., K.Y. Ekşi, JCAP 1107 (2011) 020. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

44 Equation of State 3: MS α 9 =-2.0 α 9 =-1.0 α 9 = 0.0 α 9 = 1.0 α 9 = 2.0 M * / M O R < R S EoS = MS1 The maximum mass for MS1 [Müller & Serot, 1996] satisfies the observed mass of PSR J only for α 9 < 2 though it moves away from the M-R constraint of [Özel et al., 2010] for such low values of α R * (km) S. Arapoglu, C.D., K.Y. Ekşi, JCAP 1107 (2011) 020. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

45 Modified gravity and magnetic field effects M.-K. Cheoun, C.D., C. Güngör, V. Keleş, C. Ryu, T. Kajino, G. Mathew, [arxiv: ]: We consider the combination of strong magnetic fields and modified gravity. By way of motivation we note that in the five dimensional unification of gravity and electromagnetism the Kaluza Klein action expands into: R R a F 2, where R(R) is the scalar curvature in five (four) dimensions, F is the four dimensional electromagnetic field strength, a relates to (the square of) the length scale of the extra dimension. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

46 Modified gravity and magnetic field effects M.-K. Cheoun, C.D., C. Güngör, V. Keleş, C. Ryu, T. Kajino, G. Mathew, [arxiv: ]: We consider the combination of strong magnetic fields and modified gravity. By way of motivation we note that in the five dimensional unification of gravity and electromagnetism the Kaluza Klein action expands into: R R a F 2, where R(R) is the scalar curvature in five (four) dimensions, F is the four dimensional electromagnetic field strength, a relates to (the square of) the length scale of the extra dimension. In the case of α 9 > 0, effects by the modified gravity can be compensated by those of a strong magnetic field. However, in the case of α 9 < 0 some equations of state which were not viable for neutron stars in the case of general relativity become viable again. M * / M R * (km) α 9 =-2 α 9 =-1 α 9 = 0 α 9 = 1 α 9 = 2 nob-gr EoS = 3b4 b-nph Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

47 Conclusions for R + αr 2 theory Main points of this study: 1 Field equations of even the simplest modification of GR are complicated. In perturbative approach it becomes possible to derive the modified TOV equations. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

48 Conclusions for R + αr 2 theory Main points of this study: 1 Field equations of even the simplest modification of GR are complicated. In perturbative approach it becomes possible to derive the modified TOV equations. 2 The result of (Özel et al., 2010) and measured mass of PSR J (Demorest et al., 2010) excludes many EoSs (especially soft ones) in the framework of GR. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

49 Conclusions for R + αr 2 theory Main points of this study: 1 Field equations of even the simplest modification of GR are complicated. In perturbative approach it becomes possible to derive the modified TOV equations. 2 The result of (Özel et al., 2010) and measured mass of PSR J (Demorest et al., 2010) excludes many EoSs (especially soft ones) in the framework of GR. 3 In the f (R) = R + αr 2 gravity model, the value of α provides a new degree of freedom and for certain values of it some of the EoSs, which are excluded within the framework of GR, can now be reconciled with the observations. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

50 Conclusions for R + αr 2 theory Main points of this study: 1 Field equations of even the simplest modification of GR are complicated. In perturbative approach it becomes possible to derive the modified TOV equations. 2 The result of (Özel et al., 2010) and measured mass of PSR J (Demorest et al., 2010) excludes many EoSs (especially soft ones) in the framework of GR. 3 In the f (R) = R + αr 2 gravity model, the value of α provides a new degree of freedom and for certain values of it some of the EoSs, which are excluded within the framework of GR, can now be reconciled with the observations. 4 We found the constraint α cm 2 independent of the EoS. Presence of uncertainties in the EoS does not cloak the effect of α on the results. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

51 Conclusions for R + αr 2 theory Main points of this study: 1 Field equations of even the simplest modification of GR are complicated. In perturbative approach it becomes possible to derive the modified TOV equations. 2 The result of (Özel et al., 2010) and measured mass of PSR J (Demorest et al., 2010) excludes many EoSs (especially soft ones) in the framework of GR. 3 In the f (R) = R + αr 2 gravity model, the value of α provides a new degree of freedom and for certain values of it some of the EoSs, which are excluded within the framework of GR, can now be reconciled with the observations. 4 We found the constraint α cm 2 independent of the EoS. Presence of uncertainties in the EoS does not cloak the effect of α on the results. 5 Gravity Probe B data implies α cm 2 (Naf & Jetzer, 2010). Neutron stars are indeed useful for constraining gravity even if we do not know the EoS! Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

52 Another modification of the same order Motivation from string theory: Einstein-Hilbert modified with Gauss-Bonnet term S = d 4 x g [ R + γ(r 2 4R µν R µν + R µνρσ R µνρσ ) ] Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

53 Another modification of the same order Motivation from string theory: Einstein-Hilbert modified with Gauss-Bonnet term S = d 4 x g [ R + γ(r 2 4R µν R µν + R µνρσ R µνρσ ) ] GB term do not contribute to field equations in 4D. For the classical physics applications S = d 4 x g [ R + αr 2 + βr µν R µν] + S matter Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

54 Another modification of the same order Motivation from string theory: Einstein-Hilbert modified with Gauss-Bonnet term S = d 4 x g [ R + γ(r 2 4R µν R µν + R µνρσ R µνρσ ) ] GB term do not contribute to field equations in 4D. For the classical physics applications S = d 4 x g [ R + αr 2 + βr µν R µν] + S matter Field equations for α=0 and arbitrary β 8πT µν = G µν + β ( 1 2 g µνr ab R ab + ρ ) ρ R µν +β ( ν µ R 2R σµνα R ασ Rg ) µν We found the constraint β cm 2 independent of the EoS. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

55 A unitary gravity theory in 4 dimensions A renormalizable gravity theory in 4D S = d 4 x g [ R + αr 2 + βr µν R µν] This theory describes a massless spin-2 graviton, a massive spin-2 ghost field and a massive scalar (Stelle, 1977 & 1978). Massive spin-2 is absent in the case of β = 3α, while massive spin-2 is absent if β = 0. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

56 A unitary gravity theory in 4 dimensions A renormalizable gravity theory in 4D S = d 4 x g [ R + αr 2 + βr µν R µν] This theory describes a massless spin-2 graviton, a massive spin-2 ghost field and a massive scalar (Stelle, 1977 & 1978). Massive spin-2 is absent in the case of β = 3α, while massive spin-2 is absent if β = 0. A unitary gravity theory in 4D S = d 4 x g [ R 2Λ 1 ] 2Λ (R2 3R µν R µν ) This theory describes two massless spin-2 gravitons without any scalar (Lü & Pope, 2011). Even though it is non-renormalizable, it could be taken as a toy model for quantum gravity in 4 dimensions. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

57 Future directions 1 Look for how to implement effects of magnetic field in modified TOV equations. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

58 Future directions 1 Look for how to implement effects of magnetic field in modified TOV equations. 2 We plan to analyze if supersoft EoSs (Wen et al., 2009) can be reconciled with the observations using the modified TOV equations. Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26

59 Future directions 1 Look for how to implement effects of magnetic field in modified TOV equations. 2 We plan to analyze if supersoft EoSs (Wen et al., 2009) can be reconciled with the observations using the modified TOV equations. 3 It would be significant to understand the effect of Weyl-tensor-squared term on neutron star physics: 1 2 CµνρσC µνρσ = 1 3 R2 + R µνr µν (modulo GB term) S = d 4 x g [R 2Λ + γc µνρσc µνρσ ] + S matter Cemsinan Deliduman (MSU) NS as Laboratories for Gravity Physics CETUP* July 25, / 26