Rotating neutron stars in scalar-tensor theories of … rotating neutron stars in scalar-tensor theories of gravity Daniela Doneva University of Tübingen In collaboration with: Stoytcho

Rapidly rotating neutron stars in scalar-tensor theories of gravity

Daniela Doneva University of Tübingen

In collaboration with: Stoytcho Yazadjiev, Nikolaos Stergioulas and Kostas Kokkotas

Phys. Rev. D 88, 084060 (2013)

SFB/TR7 Video Seminar Daniela Doneva

Rapidly rotating neutron stars in scalar-tensor theories of gravity

Plan of the talk: • Introduction to scalar-tensor theories

• Scalarized static neutron stars

• Field equations and code implementation

• Results

• Astrophysical implications

• Conclusions

Slide: 2


Scalar-tensor theories of gravity - motivation

• The scalar-tensor theories are one of the most natural generalizations of the

Einstein theory of gravity.

• Their essence is in one or several scalar fields that are mediators of the

gravitational interaction in addition to the spacetime metric in the classical Einstein

theory.

• Scalar fields appear in the reduction of Kaluza-Klein theories to four dimensions,

in string theory and in higher dimensional gravity but scalar-tensor theories can be

defined completely independently.

• Scalar-tensor theories can be consider as an Einstein theory of gravity but with

variable gravitational constant.

• They fit the observational data very well.

• They are also an essential part of dark energy and dark matter models.

• 𝑓 𝑅 theories are mathematically equivalent to the scalar-tensor theories.

Slide: 3

Physical (Jordan) frame action:

Einstein frame action (much simpler):

• Coupling function:

• We set the potential to zero 𝑉 𝜑 = 0


Scalar-tensor theories of gravity – the action

Slide: 4

Do the compact objects, such as black holes and neutron stars, differ in scalar-tensor theories?


Scalar-tensor theories of gravity – compact objects

Field equation for the scalar field – nonzero trace 𝑇 acts as a source for the scalar field

• No scalar-hair conjecture: black holes in scalar-tensor theories are indistinguishable from GR. Not applicable when the trace of the energy momentum tensor is nonzero.

Possible scenarios: changed black holes described by nonlinear electrodynamics (Stefanov, Yazadjiev and Todorov (2008), Doneva at al (2010)), black holes surrounded by matter (Cardoso et al. (2013)).

Slide: 5

Two standard choices of the coupling function 𝑘(𝜑):

1. 𝑘 𝜑 = 𝑘0 • Equivalent to the Brans-Dicke theory. • Differs from GR in the weak field regime. • Neutron stars have nontrivial scalar field for every 𝑘0 ≠ 0

2. 𝑘 𝜑 = 𝛽𝜑

• Equivalent to GR in the weak field regime. • Can differ significantly when strong fields are considered. • Nonuniqueness of the neutron star solutions can exist – one solution

with trivial scalar field and one or several others with nontrivial scalar field.


Scalar-tensor theories of gravity – neutron stars

• Scalarization of neutron stars in the second class of scalar-tensor theory was considered for the first time in Damour&Esposito-Farese (1993)

• Slow rotation approximation (or order Ω) was also considered (Damour&Esposito-Farese

(1996), Sotani (2012)).

Slide: 6

Properties of the static scalarized neutron stars

The solutions with nontrivial scalar field are energetically more favorable than their GR counterpart (Harada 1997, Harada 1998).


Scalar-tensor theories of gravity – neutron stars

Slide: 7

Observational constraints: approximatelly 𝑘0 < 0.004 and 𝛽 > −4.8 (Damour & Esposito-Farese

(1996,1998), Will (2006), Freire at al (2012))


Scalar-tensor theories of gravity - observations

Freire at al (2012)

𝑘0

Slide: 8


Field equations

A general ansatz for a stationary and axisymmetric metric:

The reduced field equations for the metric functions:

Slide: 9


Field equations

The reduced field equation for the metric function 𝛼:

Equations for the scalar field and for hydrostationary equilibrium:

Slide: 10

• Polytropic EOS with N = 0.7463 and K = 1186 (EoS II)

• Global physical characteristics of the star:


Neutron-star characteristics

Slide: 11

• A modification of the RNS code by N. Stergioulas (Stergiulas & Friedman (1995)) is developed.

• The code uses the KEH method (Komatsu, Eriguchi, Hachisu(1898)) with a modification introduced by Cook, Shapiro and Teukolski (1994).

• The field equations are presented in an integral form using appropriate Green functions.

• The code is tested successfully against several limiting cases: The GR limit The nonrotating limit The slow rotation approximation (of order Ω) The mass and the angular momentum are derived in two independent ways: by

integral formulae and by the metric asymptotics at infinity.


Numerical implementation

Slide: 12

Constant coupling function 𝒌 𝝋 = 𝒌𝟎.

Contour plot of the scalar field for 𝑘0 = 4x10−3 and 𝜀 𝑐 = 1.25 × 1015 g/cm3


Results

Nonrotating, Ω = 0 Kepler limit, Ω = Ω𝐾

Slide: 13

Coupling function 𝒌 𝝋 = 𝜷𝝋 – spontaneous scalarization, nonuniqueness of the solutions

• The effect of scalarization is much stronger for fast rotation. • Scalarized solutions exist for a much larger range of parameters than in

the static case.


Results

Slide: 14

Angular momentum and moment of inertial – could differ twice for scalarized solutions


Results

Sequences of models rotating at the Kepler limit Models with constant central energy density

Slide: 15

Stability of the solutions : The scalarized neutron stars are energetically more favorable!

Sequences of fixed angular momentum 𝐽 are plotted.


Results

Magnification of the left figure: The relative energy as a function of the rest mass:

Slide: 16

Quasi-periodic oscillations of neutron stars 1. Quasi-periodic oscillation are observed in the X-ray spectrum of compact objects such

as black holes, neutron stars and white dwarfs.

2. The X-rays are supposed to be emitted from the inner edge of an accretion disk.

3. Many models for the QPOs exist and one of the most popular is that they are connected with the epicyclic frequencies (radial and vertical).

4. Innermost stable circular orbit (ISCO) – the inner edge of the accretion disc.


Astrophysical implication

Slide: 17



Equations for the orbital frequency of a particle Ω𝑝, the radial 𝜔𝑟 and vertical 𝜔𝜃 epicyclic frequencies:

Slide: 18

Quasi-periodic oscillations of scalarized neutron stars

Doneva, Yazadjiev, Stergioulas, Kokkotas (in preparation)



The orbital frequency at ISCO The maximum radial frequency as a function of the mass

Slide: 19

Quasi-periodic oscillations of scalarized neutron stars

The orbital and the epicyclic frequencies for a single star rotating with 𝜈 = 700Hz and mass 𝑀 = 2.0𝑀⨀ (EoS APR) and 𝑀 = 2.3𝑀⨀ (EoS L)

Doneva, Yazadjiev, Stergioulas, Kokkotas (in preparation)



Slide: 20

Conclusions • The field equations governing fast rotating scalar-tensor neutron stars are obtained.

• An extension of the RNS code is developed which solves the field equation

• Fast rotating neutron stars with nontrivial scalar field are obtained in two classes of

scalar-tensor theories.

• The influence of the scalar field is much more pronounced for fast rotation.

• Scalarization exists for a wider range of parameters compared to the static case.

• Astrophysical implications are considered.


Conclusions

Slide: 21

Thank you!

Documents

Rotating neutron stars in scalar-tensor theories of … rotating neutron stars in scalar-tensor theories of gravity Daniela Doneva University of Tübingen In collaboration with: Stoytcho