The mass of dark scalar and phase space analysis of realistic models of static spherically symmetric

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1 The mass of dark scalar and phase space analysis of realistic models of static spherically symmetric objects (NS) Plamen Fiziev BLTF, JINR, Dubna and Sofia University Talk at the MG14, July 2015, Roma, Italy supported by Bulgarian Nuclear Regulatory Agency, COST Action MP1304 NewCompStar, and TCPA Foundation Plan of the talk: General remarks on DM and DE Minimal dilatonic gravity (MDG) The basic equations and BC of SSSC in MDG The dark scalar SSSO (NS with realistic EOS) in MDG Singular manifolds and the Lyapunov analysis SSSO = Static Spherically Symmetric Objects

2 The basic lesson from cosmology: GR and SPM are not enough! One may add some new content: Dark matter (DM), Dark energy (DE) One may modify GR: simplest modifications are F(R) and MDG Some combination of the above two possibilities may work? 2015 Planck results: While the need for DM and DE is strongly established, their nature and their small-scale distribution are still largely unknown. The only established part about DM is its gravitational interaction. We have, in fact, no evidence that DM has any other interaction but gravity! The DE in MDG services for description of both initial and present expansion of the Universe.

3 Most probably we need to look simultaneously and coherently for a realistic EOS and for a realistic modified gravity which are able to describe a variety of cosmological, astrophysical, gravitational and star phenomena at different scales: Planets, compact stars, withe dwarfs, normal stars, stars clusters, dwarf sphericals, galaxies, galaxy clusters, and at the scale of the whole Universe It is not excluded that these objects are related with different de Sitter vacuums, suitable for corresponding different scales and for different time epochs.

4 Minimal dilatonic gravity (MDG) NO Φ + A matter enters In GR with cosmological constant :, O Hanlon: PRL, 1972, PPF: Mod. Phys. Lett. A, 15, 1077 (2000); gr-qc/ ; PRD 67, (2003); PRD 87, (2013); PoS (FFP14) 080 (1914); PPF, K. Marinov: BAJ, 23, 1 (2015); PPF, arxiv: Gravitational factor NEW: variable Cosmological factor Λ > 0 Φ > 0 U > 0 Observed value: Λ cm 2 Very small MDG is locally equivalent to f(r).

5 Basic Equations of MDG: (PPF ) Cosmological principle respected Energy-momentum conservation respected

6 Withholding potentials: PPF: MPLA (2000); PRD (2013) No ghosts! No tachyons! MDG is consistent with Solar system experiments: PF 2000 λ C ~ 10 2 cm Comparison of the Starobinsky potentials VSt and dilatonic potential V with identical masses of the scalaron and MDG-dilaton: λ C ~ cm

7 The basic equations of SSSC in MDG PF: arxiv: arxiv: Generalized TOV equations: Non autonomous system : In GR: 3 rd order autonomous ODE In MDG: 5 th order autonomous ODE A =

8 NOVEL Quantities and EOS: PF: arxiv: arxiv: Dark energy-density and pressure: Dark matter-density and pressure: MDG analog of the harmonic oscillator V(Φ) = μ2 2 (Φ + 1 Φ 2) П = λ C Λ ~ ~ λ C = ђ / m Φ c

9 The Dark Scalar Stretching the physical domain in the phase space Φ = exp(a exp(ϕ n )-1) The dilaton 0 < Φ < ϕ = n ln(1+ a 1 ln Φ) The dark scalar a > 0 < ϕ < Use a = 1, n = 1, 3, 5,

10 r c = 0 Schematic procedure for calculations Center of the star Edge of the star Boundary of the Universe Fixed singular boundary Moving regular boundary Moving singular boundary r r U m c = 0 p c > 0 4 Eqs m p = 0 3 Eqs m tot Δ = 0 Φ c > 1 p Φc (p c, Φ c )= 2 3 ρc2 3+p 2 3 ϱc2 V Φ c Non autonomous Φ > 1 p Φ Non autonomous Φ U = 1 p ΦU Two specific MDG relations One parametric ( p c ) family of SSSC as in GR and the Newton gravity!

11 The Border of the MDG-Kottler-Weyl-like Universe Kottler (1918)-Weyl (1919): ds 2 = (1-2m r - Λ 3 r2 )dt 2 - (1-2m r - Λ 3 r2 ) 1 dr 2 - r 2 dω 2 m = const (Schwarzschild-de Sitter Universe) The MDG-One-SSSO-Universe: (a sckech!) Cosmological Horizon: g tt = 0, g rr =, g tt g rr = - c 2 λ C ~ km Stop! dsv: Φ = 1 R U (4) = 4Λ - 2/rU 2 R U (3) = 2 Λ r SSSO Using a sophisticated shooting method for the BVP

12 Some new MDG-results for MEOS AMP1: arxiv: Nuclear densitiy g/cm 3 ρ(r) g/cm 3 Fe 56 densities 6.49 g/cm 3 Neutron star Dark domain NS Dark domain

13 Some new MDG-results for MEOS AMP1: arxiv: weaker gravity weaker gravity p Φ (r) p Φ (r)

14 Some new MDG-results for MEOS AMP1: arxiv: μ critical = 1 / Π critical A new phenomenon: Shrinkage of the domain of initial conditions approaching the bifurcation point:

15 Some new MDG-results for MEOS AMP1: arxiv: GR Shrinkage of the domain μ critical Infinite m φ GR MDG Small m φ MDG GR

16 Compactness of MDG-NS for MEOS AMP1 and for different masses of the dark filed: arxiv: GR Shrinkage of the domain μ critical Infinite m φ GR Small m φ MDG

17 Cosmological units : ~ Regularization parameter ( ) = d dτ Autonomos system in

18 Singular manifolds

19 The maps defined by

20 The geometrical radial distance Coordinate independent definition of singularities The Lyapunov Function V Φ; iμ = V Φ; iμ = + 0

21 0

22 Basic results: MDG model, being locally equivalent to f(r), is shown to be much more physically intuitive. The realistic models of different SSSO are possible. Numerical difficulties are surmounted introducing the dark scalar ϕ = n ln(1+ a 1 ln Φ). The dark domain around SSSO is proven to exist and New phenomenon: a shrinkage of the physical domain shows the existence of bifurcation point, which depends on the mass of the dark scalar. All singular sub-manifolds of different dimension in the phase space are discovered. The Lyapunov analysis of trajectories is described. 16/06/2015, Budapest

23 More efforts are needed to know more about the influence of dark matter and dark energy on the SSSC Thank you!