.. Neutron Stars in Alternative Theories of Gravity Jutta Kunz Institute of Physics CvO University Oldenburg NewCompStar School 2017 Neutron stars: theory, observations and gravitational waves emission Sofia, 11. 15. September 2017 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 1 / 27
. Outline Outline. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 2 / 27
. Outline Outline. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 2 / 27
. Outline Outline. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 2 / 27
. Outline Outline. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 2 / 27
. Outline Outline. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 2 / 27
. Outline Motivation. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
. Motivation Motivation General Relativity G µν + Λg µν = 8πT µν In four spacetime dimensions the only divergence-free symmetric rank-2 tensor constructed solely from the metric and its derivatives up to second differential order, is the Einstein tensor plus a cosmological term. Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
. Motivation Motivation General Relativity Compatible with all solar system observations Compatible with all astrophysical observations New Observations in the strong gravity regime? Black holes Neutron stars Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
. Motivation Motivation General Relativity Incompatibility with Quantum Mechanics Singularities Dark Matter, Dark Energy Future Observations? Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
Motivation. Motivation GR or Alternative Theories of Gravity Scalar-tensor theories f (R) theories Higher curvature theories Massive gravity... Jutta Kunz (Universita t Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
. Motivation Motivation GR or Alternative Theories of Gravity Scalar-tensor theories f(r) theories Higher curvature theories Massive gravity... Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
. Motivation Motivation Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 3 / 27
. Outline Neutron Stars in General Relativity. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 4 / 27
Neutron Stars in General Relativity. Neutron Stars in General Relativity Demorest et al., Nature 476, 1081 (2010) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 4 / 27
Neutron Stars in General Relativity. Neutron Stars in General Relativity Cipolletta et al., arxiv:1506.05926: RMF 2.5 2.0 Static sequence Keplerian sequence f =50 Hz f =200 Hz f =300 Hz f =500 Hz f =716 Hz M [M ] 1.5 1.0 0.5 12 14 16 18 20 22 24 26 R eq [km] Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 4 / 27
Neutron Stars in General Relativity. Neutron Stars in General Relativity Cipolletta et al., arxiv:1506.05926: RMF 2.5 2.0 M [M ] 1.5 1.0 0.5 Static sequence Keplerian sequence Secular Instability f =50 Hz f =200 Hz f =300 Hz f =500 Hz f =716 Hz 14.4 14.6 14.8 15.0 15.2 log(ε c /c 2 [g cm 3 ]) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 4 / 27
Neutron Stars in General Relativity. Neutron Stars in General Relativity Cipolletta et al., arxiv:1506.05926: RMF energy density contours: ϵ c = 10 15 g/cm 3 static rotating Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 4 / 27
Neutron Stars in General Relativity. Neutron Stars in General Relativity Cipolletta et al., arxiv:1506.05926: RMF 5 4 Keplerian sequence f =50 Hz f =200 Hz f =300 Hz f =500 Hz f =716 Hz I [10 45 g cm 2 ] 3 2 1 0 0.5 1.0 1.5 2.0 2.5 M [M ] Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 4 / 27
. Outline Neutron Stars in Scalar-Tensor Theories. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 5 / 27
. Outline Neutron Stars in Scalar-Tensor Theories STT. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 5 / 27
Neutron Stars in Scalar-Tensor Theories. Scalar-Tensor Theories STT action: Jordan frame 1 S = d 4 x g 16πG G : gravitational constant R: Ricci scalar with respect to g µν Φ: graviational scalar field S m : matter action Ψ m : matter fields [ F (Φ) R ] Z(Φ) g µν µ Φ ν Φ 2U(Φ) + S m [Ψ m ; g µν ] Φ does not couple directly to Ψ m : weak equivalence principle is satisfied Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 6 / 27
Neutron Stars in Scalar-Tensor Theories. Scalar-Tensor Theories STT action: Jordan frame 1 S = d 4 x g 16πG [ F (Φ) R ] Z(Φ) g µν µ Φ ν Φ 2U(Φ) + S m [Ψ m ; g µν ] functions F (Φ) and Z(Φ), and potential function U(Φ) physical restrictions positivity of the graviton energy: F (Φ) > 0 positivity of the scalar field kinetic energy: ( ) 2 d 2F (Φ)Z(Φ) + 3 dφ F (Φ) 0 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 6 / 27
Neutron Stars in Scalar-Tensor Theories. Scalar-Tensor Theories STT transformation to Einstein frame action Einstein in frame ( ) 2 dφ = 3 dφ 4 ( d ln(f (Φ)) dφ ) 2 + Z(Φ) 2F (Φ) 1 S = 16πG d 4 x g [R 2g µν µ φ ν φ 4V (φ)] + S m [Ψ m ; A 2 (φ)g µν ] relations between the Jordan frame functions F (Φ) and U(Φ) and the Einstein frame functions A(φ) and V (φ) A(φ) = F 1/2 (Φ), 2V (φ) = U(Φ)F 2 (Φ) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 6 / 27
Neutron Stars in Scalar-Tensor Theories. Scalar-Tensor Theories STT equations in the Einstein frame R µν 1 2 g µνr = 2 µ φ ν φ g µν g αβ α φ β φ 2V (φ)g µν + 8πG T µν µ d ln(a(φ)) µ φ = 4πG T µ dv (φ) µ + dφ dφ energy momentum tensor: Jordan frame T µν = ( ε + p)ũ µ ũ ν + p g µν energy momentum tensor: Einstein frame T µν = (ε + p)u µ u ν + pg µν A 2 Tµν ε = A 4 (φ) ε, p = A 4 (φ) p, u µ = A 1 (φ)ũ µ Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 6 / 27
Neutron Stars in Scalar-Tensor Theories. Scalar-Tensor Theories STT Brans Dicke Theory S = 1 d 4 x [ g 16π ϕr ω(ϕ) ϕ ] gµν ( µ ϕ) ( ν ϕ) U(ϕ) + S M [Ψ, g µν ] relation between Jordan-frame and Einstein-frame quantites ϕ = A 2 (φ), 3 + 2ω(ϕ) = α(φ) 2 α(φ) d(ln A(φ))/dφ α(φ) = α 0 =constant, i.e., ω(ϕ) =constant observational bound: ω > 40000 (Cassini-Huygens) limit: ω GR Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 6 / 27
Neutron Stars in Scalar-Tensor Theories. Observational Bounds STT Freire et al. 1205.1450 0 observational constraints on STT parameters ln A(φ) = ln A(φ 0 ) +α 0 (φ φ 0 ) + 1 2 β 0(φ φ 0 ) 2 Cassini LLR 10 100 10 10 B1534+12 SEP J0737 3039 B1913+16 LLR J1141 6545 J1738+0333 m φ = 0 10 6 4 2 0 2 4 6 0 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 7 / 27
Neutron Stars in Scalar-Tensor Theories. Observational Bounds STT Antoniadis et al. 1304.6875 observational constraints on STT parameters α 0 = 10 4 5 β 0 4 top to bottom steps 0.1 m φ = 0 β 0 4.5 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 7 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in STT Theories STT constraints on STT parameters: negligible effect on neutron stars? non-perturbative effects in strong fields: spontaneous scalarization! Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 8 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in STT Theories STT Doneva et al. 1309.0605 z/r e ln A(φ) = α 0 φ, α 0 < 4 10 3 4 3 2 1-1.2E-03-1.0E-03-8.9E-04-7.4E-04-5.9E-04-4.5E-04-3.0E-04-1.5E-04 0.0E+00 z/r e 4 3 2 1-1.3E-03-1.2E-03-1.0E-03-8.4E-04-6.7E-04-5.1E-04-3.4E-04-1.7E-04 0.0E+00 0 0 1 2 3 4 0 0 1 2 3 4 x/r e static x/r e Kepler limit scalar field φ, neutron star surface (black dashed) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 8 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in STT Theories STT Esposito-Farese (lecture) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 8 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars: Spontaneous Scalarization for example Altaha Motahar et al. 1707.05280 M (M O ) 2.5 2 1.5 mass radius relation BHZBM H4 GNH3 GR A 1, β 1 A 1, β 2 A 1 (φ) = e 1 2 βφ2 A 2, β 1 A 2, β 2 1 A 2 (φ) = cosh( βφ) 2.5 β 1 = 4.8 M (M O ) 2 1.5 2.5 WCS1 WCS2 polytrope β 2 = 4.5 Damour, Esposito-Farese M (M O ) 2 1.5 SLy WSPHS1 APR4 WSPHS2 WSPHS3 A 1 (φ) = e 1 2 βφ2 A 3 (φ) = cos( βφ) Jutta2.5 Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars: Spontaneous Scalarization M (M O ) 2.5 2 STT BHZBM GR A 1, β 1 A 1, β 2 A 2, β 1 A 2, β 2 1.5 H4 GNH3 2.5 M (M O ) 2 1.5 WCS1 WCS2 polytrope 2.5 M (M O ) 2 WSPHS1 WSPHS2 1.5 SLy APR4 WSPHS3 2.5 M (M O ) 2 1.5 ALF2 ALF4 BS4 11 12 13 14 11 12 13 14 11 12 13 14 R s (Km) R s (Km) R s (Km) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars: Spontaneous Scalarization ω A 0.01 0 H4 BHZBM STT GR A 1, β 1 A 1, β 2 A 2, β 1 A 2, β 2-0.01 GNH3 0.01 ω A 0 WCS1 polytrope ω A -0.01 0.01 0 WCS2 scalar charge ω A φ(r) = ω A r +... -0.01 SLy WSPHS1 APR4 WSPHS2 WSPHS3 0.01 ω A 0 ALF2 ALF4 BS4-0.01 1.4 2 2.6 1.4 2.0 2.6 1.4 2.0 2.6 M (M O ) M (M O ) M (M O ) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars: Spontaneous Scalarization for example Altaha Motahar et al. 1707.05280 β cr -4.343-4.345-4.347 fit f(x) polytrope SLy APR4 BHZBM GNH3 H4 WCS1 WCS2 ALF2 ALF4 WSPHS3 WSPHS1 WSPHS2 f(x) = c 0 +c 1 (M/R s ) -1 +c 4 (M/R s ) -4 ω A 0.02 0.01 0-0.01-4.349 0.21 0.22 0.23 0.24 0.25 M/R s onset of scalarization β crit vs compactness M/R fit function -0.02-0.03 A=e βϕ2 /2 β=-4.8 polytrope SLy APR4 BHZBM WCS1 WCS2 GNH3 H4 WSPHS3 ALF2 ALF4 WSPHS1 WSPHS2 BS4 0.18 0.21 0.24 0.27 M/R s scalarized neutron stars scalar charge ω A vs compactness M/R Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars: Spontaneous Scalarization I (10 45 g cm 2 ) 4.5 H4 3 1.5 BHZBM STT GNH3 GR A 1, β 1 A 1, β 2 A 2, β 1 A 2, β 2 I (10 45 g cm 2 ) 4.5 WCS1 3 WCS2 polytrope 1.5 I (10 45 g cm 2 ) 4.5 WSPHS1 3 1.5 SLy WSPHS2 APR4 WSPHS3 I (10 45 g cm 2 ) 4.5 3 ALF2 ALF4 BS4 1.5 1.4 2 2.6 1.4 2.0 2.6 1.4 2.0 2.6 M (M O ) M (M O ) M (M O ) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars: Spontaneous Scalarization Doneva et al. 1309.0605 M/M < ln A(φ) = 1 2 β 0φ 2, β 0 > 4.5 3.0 2.5 2.0 1.5 1.0 = 0 = -4.2 = -4.5 = -4.8 10 12 14 16 18 20 22 R e [km] mass radius M/M < 3.0 2.5 2.0 1.5 1.0 0.5 0.0 = 0 = -4.2 = -4.5 = -4.8 1x10 15 2x10 15 3x10 15 4x10 15 ~ 2 c / c [ g / cm 3 ] mass energy density spontaneous scalarization: static and Kepler limit Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars: Spontaneous Scalarization Doneva et al. 1309.0605 ln A(φ) = 1 2 β 0φ 2, β 0 > 4.5 0.0 c -0.1-0.2 = 0 = -4.2 = -4.5 = -4.8-0.3 0.0 2.0x10 15 4.0x10 15 ~ 2 / c [ g / cm 3 ] central value of the scalar field vs central energy density static sequences (solid lines) Kepler limit sequences (dotted lines) c Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars: Spontaneous Scalarization Staykov et al. 1602.00504 ln A(φ) = 1 2 β 0φ 2 0.5 I/(MR 2 ) 0.4 0.3 STT, j = 0.2 GR, j = 0.2 STT, j = 0.4 GR, j = 0.4 STT, j = 0.6 GR, j = 0.6 GR fit, slow rot. 1 - I GR /I fitgr 1 - I STT /I fitstt 0.1 0.01 1E-3 1E-4 0.1 0.01 1E-3 1E-4 0.10 0.15 0.20 0.25 0.30 0.35 M/R scaled moment of inertia vs compactness Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 9 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in massive STT Theories STT Alsing et al. 1112.4903 massive variant of the Brans-Dicke theory bounds on Brans-Dicke coupling parameter ω BD and scalar mass m s Cassini measurements of the Shapiro time delay ω BD > 40000 for scalar masses m s < 2.5 10 20 ev Compton wavelengths λ s = h/(m sc) > 5 10 10 km Lunar Laser Ranging (LLR) experiments ω BD > 1000 for m s < 2.5 10 20 ev white dwarf-neutron star binary PSR J1012+5307 ω BD > 1250 for m s < 10 20 ev Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 10 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in massive STT Theories STT Alsing et al. 1112.4903 10 5 Lower bound on (ω BD +3/2) 10 4 10 3 10 2 10 1 10 0 10-1 Cassini J1141-6545 J1012+5307 LLR 10-2 10-21 10-20 10-19 10-18 10-17 10-16 10-15 m s (ev) bounds on Brans-Dicke coupling parameter ω BD and scalar mass m s vertical lines: 1 AU (solid), orbital radii (dashed) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 10 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in massive STT Theories STT Alsing et al. 1112.4903 10 5 Lower bound on (ω BD +3/2) 10 4 10 3 10 2 10 1 10 0 10-1 Cassini J1141-6545 J1012+5307 LLR 10-2 10-21 10-20 10-19 10-18 10-17 10-16 10-15 m s (ev) bounds on Brans-Dicke coupling parameter ω BD and scalar mass m s consequences: scalar field mass m φ : finite range of the scalar field of order Compton wavelength λ φ no/less stringent constraints for observations involving distances > λ φ Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 10 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in massive STT Theories STT Doneva et al. 1602.04766 Brans-Dicke theory: A(φ) = e α0φ Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 10 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in massive STT Theories STT Doneva et al. 1602.04766 slow rotation Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 10 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars in massive STT Theories Ramazoglu et al. 1601.07475 spontaneous scalarization bounds on scalar mass m φ : bound on the λ φ (m φ ): binary system (with smallest) orbital separation 10 9 m = m φ 10 16 ev characteristic length scale of star smaller than λ φ m φ 10 9 ev bounds on β white dwarfs: no scalarization 10 3 < β < 3 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 11 / 27
Neutron Stars in Scalar-Tensor Theories. Neutron Stars in massive STT Theories STT Ramazoglu et al. 1601.07475 spontaneous scalarization upper row: m ϕ = 1.6 10 13 ev middle row: m ϕ = 4.8 10 13 lower row: m ϕ = 1.6 10 12 ev left column: β = 4.5 middle column: β = 6 right column: β = 10 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 11 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars in massive STT Theories Doneva et al. 1602.04766, 1607.03299 static, scalarization: A(φ) = e 1 2 βφ2 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 11 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars in massive STT Theories Doneva et al. 1602.04766, 1607.03299 slow rotation, scalarization: A(φ) = e 1 2 βφ2 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 11 / 27
Neutron Stars in Scalar-Tensor Theories STT. Neutron Stars in massive STT Theories Doneva et al. 1602.04766, 1607.03299 rapid rotation, scalarization: A(φ) = e 1 2 βφ2 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 11 / 27
. Outline Neutron Stars in Scalar-Tensor Theories Horndeski. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 12 / 27
. Horndeski gravity Neutron Stars in Scalar-Tensor Theories Horndeski most general scalar-tensor theory with second-order field equations and one scalar field S = d 4 x { g K(ϕ, X) G 3 (ϕ, X) ϕ + G 4 (ϕ, X)R + G 4,X (ϕ, X) [ ( ϕ) 2 ( µ ν ϕ)( µ ν ϕ) ] + G 5 (ϕ, X)G µν µ ν ϕ G 5,X(ϕ, X) [ ( ϕ) 3 3 ϕ( µ ν ϕ)( µ ν ϕ) 6 + 2( µ ν ϕ)( µ σ ϕ)( ν σ ϕ) ]} K and G i s (i = 1... 5): functions of the scalar field ϕ and of its kinetic term X = 1/2 µ ϕ µ ϕ G i,x : derivatives of G i with respect to X Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27
Neutron Stars in Scalar-Tensor Theories Horndeski. Horndeski gravity Charmousis et al. 1106.2000, 1112.4866 cosmological considerations subsector: Fab Four ) ( S = d4 x g LG [gµν, ϕ] + LM [gµν, Ψ] LG [gµν, ϕ] = Ljohn + Lpaul + Lgeorge + Lringo Jutta Kunz (Universita t Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27
. Horndeski gravity Neutron Stars in Scalar-Tensor Theories Horndeski Charmousis et al. 1106.2000, 1112.4866 cosmological considerations subsector: Fab Four S = d 4 x ( ) g L G [g µν, ϕ] + L M [g µν, Ψ] L G [g µν, ϕ] = L george + L ringo + L john + L paul special cases of Horndeski gravity general relativity ( George ) Einstein-dilaton-Gauss-Bonnet gravity ( Ringo ) theories with a nonminimal coupling with the Einstein tensor ( John ) theories involving the double-dual of the Riemann tensor ( Paul ) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27
µ µ µ µ Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27. Horndeski gravity Neutron Stars in Scalar-Tensor Theories Horndeski Charmousis et al. 1106.2000, 1112.4866 Fab Four Gauss-Bonnet term L george = V george (ϕ)r L ringo = V ringo (ϕ)r GB L john = V john (ϕ)g µν µ ϕ ν ϕ L paul = V paul (ϕ)p µναβ µ ϕ α ϕ ν β ϕ R GB R αβµν R αβµν 4R µν R µν + R 2 potentials V george (ϕ), V ringo (ϕ), V john (ϕ), and V paul (ϕ) double-dual of the Riemann tensor P µν αβ 1 4 δµνγδ σλαβ Rσλ γδ
. Horndeski gravity Neutron Stars in Scalar-Tensor Theories Horndeski Maselli et al. 1603.04876 John S G = = d 4 x g(l george + L john ) d 4 x [ g κr + 1 ] 2 ηgµν µ ϕ ν ϕ Paul L = L george + L paul = R 1 3 αp µναβ µ ϕ α ϕ ν β ϕ Paul term does not allow for physically well-behaved compact object solutions Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27
. Horndeski gravity Neutron Stars in Scalar-Tensor Theories Horndeski Maselli et al. 1603.04876 John mass radius: scalar charge q, dependence on q 2 η (constraints) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27
. Horndeski gravity Neutron Stars in Scalar-Tensor Theories Horndeski Maselli et al. 1603.04876 John moment of inertia mass: slow rotation Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 13 / 27
. Outline Neutron Stars in Scalar-Tensor Theories f(r). 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 14 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) Theories Motivation: Dark Energy action in f(r) theories of gravity S = 1 d 4 x gf(r) + S matter (g µν, χ) 16πG free of tachyonic instabilities and ghosts d 2 f/dr 2 0, df/dr > 0 mathematically equivalent to STT S = 1 d 4 x g [ΦR U(Φ)] + S matter (g µν, χ) 16πG Φ = df(r) dr, U(Φ) = R df dr f(r) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 15 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) Theories Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 Einstein frame with metric g µν conformal transformation action S = 1 16πG g µν = Φg µν d 4 x g [R 2g µν µ φ ν φ V (φ)] + S matter (e 2 3 φ g µν, χ) new scalar field φ Einstein frame potential V (φ) φ = 3 2 ln Φ V (φ) = A 4 (φ)u(φ(φ)) A 2 (φ) = Φ 1 (φ) = e 2 3 φ Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 15 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 R-squared gravity inequalities f(r) = R + ar 2 a 0, 1 + 2aR 0 gravitational scalar field and potential Φ = df(r) dr, U(Φ) = R df dr f(r) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 M/M < 2.5 2.0 1.5 1.0 0.5 EOS SLy4 GR a=0.3 a=1 a=10 a=10 2 a=10 4 0.0 9 10 11 12 13 14 15 R S [km] M/M < 2.5 2.0 1.5 1.0 0.5 0.0 EOS APR4 GR a=0.3 a=1 a=10 a=10 2 a=10 4 8 10 12 14 R S [km] mass radius relation SLy4 APR4 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 non-perturbative static results:...the differences between the R-squared gravity and general relativity are comparable with the uncertainties in the nuclear matter equations of state. That is why the current observations of the neutron star masses and radii alone cannot put constraints on the value of the parameters a, unless the equation of state is better constrained in the future. Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 I 80 60 40 APR4 GR a=0.3 a=1 a=10 a=10 2 a=10 4 SLy4 SQSB60 20 0.5 1.0 1.5 2.0 M/M < 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 M/M < M/M < slow rotation: moment of inertia mass relation APR4 SLy4 SQSB60 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 non-perturbative slowly rotating results:...the neutron star moment of inertia can be up to 30% larger compared to the corresponding general relativistic models. This is much higher than the change in the maximum mass induced by R-squared gravity and is beyond the EOS uncertainty. In this way the future observations of the moment of inertia of compact stars could allow us to distinguish between general relativity and f(r) gravity, and more generally to test the strong field regime of gravity. Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 3.0 M/M 2.5 2.0 1.5 static solutions at the Kepler frequency EOS APR4 GR a = 1 a = 10 a = 10 2 a = 10 4 static solutions at the Kepler frequency EOS SLy4 1.0 0.5 10 12 14 16 18 20 10 12 14 16 18 20 R e Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27 R e
Neutron Stars in Scalar-Tensor Theories f(r). Neutron Stars in f(r) = R + ar 2 Theory Yazadjiev et al. 1402.4469, 1407.2180, 1501.04591 I 150 100 50 0 EOS APR4 GR a = 1 a = 10 a = 10 2 a = 10 4 at the Kepler frequency static solutions 1 2 3 M/M EOS SLy4 at the Kepler frequency static solutions 1 2 3 M/M Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 16 / 27
. Outline Neutron Stars in Quadratic Gravity. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 17 / 27
Neutron Stars in Quadratic Gravity. Quadratic Gravity Curvature invariants R 2, Rµν 2, Rµνρσ 2, RR Rµν 2 R µν R µν Kretschmann scalar Rµνρσ 2 R µνρσ R µνρσ Pontryagin/Chern-Simons scalar RR 1 2 R µνρσϵ νµλκ R ρσ λκ Levi-Civita tensor ϵ µνρσ Gauss-Bonnet scalar R 2 GB R 2 4R 2 µν + R 2 µνρσ Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 17 / 27
Neutron Stars in Quadratic Gravity. Quadratic Gravity most generic class of four-dimensional theories: all quadratic algebraic curvature invariants coupled to a single scalar field S = 1 gd x[ 4 R 2 µ ϕ µ ϕ V (ϕ) 16π ] + f 1 (ϕ)r 2 + f 2 (ϕ)r µν R µν + f 3 (ϕ)r µνρσ R µνρσ + f 4 (ϕ) RR + S mat [Ψ, γ(ϕ)g µν ], coupling functions f i, potential V generic properties weak equivalence principle higher-order field equations appearance of ghosts Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 17 / 27
. Outline Neutron Stars in Quadratic Gravity EGBd Theory. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 18 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Einstein-Gauss-Bonnet-Dilaton Theory String Theory unification of all fundamental interactions dimensional reduction to 4 spacetime dimensions: low energy effective theories additional fields dilaton axion Maxwell fields Yang-Mills fields... higher order curvature corrections Gauss-Bonnet term...... Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 19 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Einstein-Gauss-Bonnet-Dilaton Theory Action S = 1 16π d 4 x [ g R 1 2 ( µϕ) 2 + α ] 4 e γϕ RGB 2 Gauss-Bonnet term: quadratic in the curvature R 2 GB = R µνρσ R µνρσ 4R µν R µν + R 2 α Gauss-Bonnet coupling constant γ dilaton coupling constant (γ = 1) In 4 spacetime dimensions the coupling to the dilaton is needed. The resulting set of equations of motion are of second order. Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 20 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Einstein-Gauss-Bonnet-Dilaton Theory modified Einstein equations G µν = 1 2 T eff µν where Tµν eff = T µν (ϕ) α 2 e γϕ T µν (GBd) T (ϕ) µν = µ ϕ ν ϕ 1 2 g µν λ ϕ λ ϕ dilaton equation T (GBd) µν = H µν + 4 ( γ 2 ρ ϕ σ ϕ γ ρ σ ϕ ) P µρνσ 2 ϕ = αγ 4 e γϕ R 2 GB Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 20 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Einstein-Gauss-Bonnet-Dilaton Theory consequences scalar hair : dilaton hair negative energy density bounds on α (γ = 1) observational Shapiro time delay α 10 13 cm BH low-mass X-ray binaries α 3.8 10 5 cm theoretical/observational lower bound on BH mass α 0.691 M 2 Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 20 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Einstein-Gauss-Bonnet-Dilaton Theory constraints from black holes 10 α L 10 α (10 5 cm) 8 6 4 γα=3.8 10 5 cm BH 1 BH 2 BH 3 GRO J0422+32 α (10 5 cm) 8 6 4 α L γα=3.8 10 5 cm BH 1 BH 2 BH 3 GRO J0422+32 2 regular dilatonic black holes, γ=1 0 0 5 10 15 20 25 M (M O ) 2 regular dilatonic black holes, γ=3 0 0 5 10 15 20 25 M (M O ) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 20 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Einstein-Gauss-Bonnet-Dilaton Theory different names: EdGB EGBd degb... different notations, different units α in cm 2 α in M 2 α dimensionless... Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 20 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Compact Stars in EGBd Theory S = 1 16π d 4 x g [R 12 ( µϕ) 2 + α4 ] e γϕ R 2GB + L matt α Gauss=Bonnet coupling constant dilaton coupling constant (γ = 1, β = 2) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 21 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Compact Stars in EGBd Theory Pani et al. 1109.0928 static neutron stars with APR EoS: dependence on α and β branches end expansion around origin: square roots reality condition: condition on αβ, maximum central density Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 21 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Compact Stars in EGBd Theory Pani et al. 1109.0928 perturbative result: moment of inertia I = J/Ω for slow rotation Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 21 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Rapidly Rotating Neutron Stars in EGBd Theory Kleihaus et al. 1601.05583 (α = 1 1 12 α L) EOS FPS EOS DI-II 2.2 2 1.8 α=0.0 α=1.0 α=2.0 EOS: FPS 2.3 2.2 2.1 2 α=0.0 α=1.0 α=2.0 EOS: DI-II M[M ] 1.6 M[M ] 1.9 1.8 static 1.4 1.2 static 1.7 1.6 1.5 1 8 9 10 11 12 13 14 15 16 17 R e [km] 1.4 10 12 14 16 18 20 R e [km] domain of existence: static, secular instability, Kepler limit Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 22 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Rapidly Rotating Neutron Stars in EGBd Theory Kleihaus et al. 1601.05583 (α = 1 1 12 α L) EOS DI-II EOS DI-II 2.3 EOS: DI-II 4 EOS: DI-II 2.2 2.1 3.5 α=1.0 M[M ] 2 1.9 1.8 1.7 1.6 1.5 α=0.0 α=1.0 α=2.0 Ω=0.020 Ω=0.030 Ω=0.038 1.4 0.5 1 1.5 2 2.5 3 ε c /c 2 [10 15 g/cm 3 ] I/10 45 [g cm 2 ] 3 α=2.0 2.5 Ω=0.020 Ω=0.030 Ω=0.038 2 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 M[M ] mass energy density moment of inertia mass Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 22 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Rapidly Rotating Neutron Stars in EGBd Theory Kleihaus et al. 1601.05583 (α = 1 1 12 α L) EOS DI-II EOS DI-II 0 35 EOS: DI-II -0.005-0.01 α=1.0 30 25 α=2.0 α=1.0 q/r 0-0.015-0.02 Ω=0.020 α=2.0-0.025 Ω=0.030 Ω=0.038 EOS: DI-II -0.03 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 M[M ] Q[M km 2 ] 20 15 10 5 Ω=0.020 Ω=0.030 Ω=0.038 0 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 M[M ] dilaton charge mass quadrupole moment mass Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 22 / 27
Neutron Stars in Quadratic Gravity EGBd Theory. Rapidly Rotating Neutron Stars in EGBd Theory Kleihaus et al. 1601.05583 (α = 1 1 12 α L) EOS DI-II EOS DI-II 20 15 10 5 EOS: DI-II α=1.0 (d) 15 10 5 0.5 0.45 0.4 0.35 0.3 0 0 0.25-5 -10-15 M=2.08 M, R e =12.3 km M=1.40 M, R e =19.8 km -20-20 -15-10 -5 0 5 10 15 20-5 -10-15 -15-10 -5 0 5 10 15 0.2 0.15 0.1 0.05 0 isometric embedding energy density (10 15 g/cm 3 ) Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 22 / 27
. EGBd Wormholes Neutron Stars in Quadratic Gravity EGBd Theory Kanti et al. 1108.3003, 1111.4049 acceleration of a traveler at the throat? g : acceleration of gravity at the surface of the earth acceleration on the order of g : throat radius on the order of (10 100) light-years Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 23 / 27
. Outline Neutron Stars in Quadratic Gravity dcs Theory. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 24 / 27
Neutron Stars in Quadratic Gravity dcs Theory. Neutron Stars in Chern-Simons Gravity action two cases dynamical: S = 1 gd 4 x [R 2 µ ϕ µ ϕ V (ϕ) + α CS ϕ RR] 16π scalar true dynamical degree of freedom dcs gravity nondynamical: scalar kinetic term absent a spherically symmetric solution of GR is also a solution of dcs gravity corrections in the presence of a parity-odd source such as rotation bound: Gravity Probe B αcs < O(10 13 )cm Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 25 / 27
Neutron Stars in Quadratic Gravity dcs Theory. Neutron Stars in Chern-Simons Gravity Yagi et al. 1302.1918 1st order α CS 2nd order J Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 25 / 27
Neutron Stars in Quadratic Gravity dcs Theory. Neutron Stars in Chern-Simons Gravity Yagi et al. 1302.1918 CS correction to quadrupole moment CS correction to angular momentum Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 25 / 27
. Outline Outlook. 1 Motivation. 2 Neutron Stars in General Relativity. 3 Neutron Stars in Scalar-Tensor Theories STT Horndeski f(r). 4 Neutron Stars in Quadratic Gravity EGBd Theory dcs Theory. 5 Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 26 / 27
. Outlook Outlook Jutta Kunz (Universität Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 26 / 27
Outlook. Thanks THANKS! Jutta Kunz (Universita t Oldenburg) Neutron Stars in Alternative Theories of Gravity NewCompStar School Sofia 2017 27 / 27