Equation of Stateof Neutron Star

with Junction Condition Approachin Starobinsky Model

Workshop on Dark Physics of the Universe National Center for Theoretical Sciences

Dec. 20th, 2015PhD student: Wei-Xiang Feng

Advisor: Prof. Chao-Qiang GengNTHU

Outline• Introduction

• The Coupled Ordinary Equations

• Junction & Boundary Conditions

• Numerical Results

• Buchdahl Stability Bound

• Summary

Introduction

Introduction： f(R) model

• Inflation model (Starobinsky model): A. A. Starobinsky “A new type of isotropic cosmological models without singularity”.Phys. Lett. B 91, 99 (1980). A. A. Starobinsky and H-J Schmidt “On general vacuum solution of fourth-order gravity”. Class.Quant.Grav. 4 (1987)

• Neutron star (NS) as a laboratory to test f(R)-theory

• Motivation: A. Ganguly, R. Gannouji, R. Goswami, and S. Ray “Neutron stars in Starobinsky model”10.1103/PhysRevD.89.064019, arXiv:1309.3279v2 [gr-qc]

Modified Gravity Action• The modified action

• After doing variation

with and

The R2 Model• R2 model (Starobinsky model)

• Field equations:

• Trace equation:

=> Curvature relates to matter differentially rather than algebraically

Introduction : Compact Star• White dwarf (WD)=> supported by degenerate electron gas

• Neutron star (NS) => supported by degenerate neutron gas & “heavy hadron repulsive force”

• When will we consider the relativistic effect?

• We can approximate the density by

• For WD,

• For NS,

=>

• For both WD and NS are around solar mass, we can infer

• From detailed calculations, , whereas .

• In fact, we can neglect the relativistic corrections for WD, however, this effect is significant for NS.

The Coupled Ordinary Equations

Computations & Numerical set-up• Spherical symmetric metric ansatz:

• Conservation law for static perfect fluid:

• Therefore, we could replace geometric parameters with physical parameters:

with

We have to force the conservation law to be valid under f(R)-theories

• For the sake of numerical set-up, we need to express

three coupled differential equations

The coupled ODEs• After laborious calculations, the results are:

Modified TOV equation

• The most different part from GR

=> Curvature relates to matter differentially rather than algebraically

Typical units • We can obtain the typical density of the EoS from one

parameter, the neutron mass , when doing phase space integration in Fermi-Dirac function. (See Weinberg p.320)

• Or we can approximate it by nucleon density,

we choose:

• The mass is around the Solar mass and the radius can be inferred once the typical density & mass are chosen

• Then we can put our equations in dimensionless form by

Junction & Boundary Conditions

Junction conditions• Schwarzschild vacuum solution:

• Junction conditions in f(R) theories (more restrictive than GR):

Apart fromtwo more conditions are requiredWith [ ] denoting the jump across the boundary surface

Ref: “Junction conditions for F(R)-gravity, and their consequences”. Jos e M. M. Senovilla. ́ arXiv:1303.1408v2 [gr-qc]

Boundary conditions

• Two first-order ODE and One second-order ODE

• The junction conditions of our problem becomes

• Regularity conditions at the center of the star

=> 4 Boundary conditions needed

• But these conditions are somewhat redundant

• Furthermore, if the EoS is chosen such that

• We are left with

(indeed for poly-trope)

automatically from modified TOV eq.

as long as

=> Five boundary conditions!! They are not independent.

• For numerical convenience we may replace

• And then check whether are satisfied

• If we assume a poly-tropic relation of EoS

They must be restricted under our boundary conditions.

Two more parameters appear !!

with random choices of

What is the reasonable ?• Important observations on the dimensionless parameters of

our system:

• should be constrained by some multiplicative combinations of these parameters.

depends on the system• At first sight, the derivative of the mass function

looks very different from usual definition

• But if we express the mass function equivalently by

Then exactly!!

• Put in dimensionless form

• Second derivative of the Ricci scalar seems problematic

as !!

• We can resolve it by the following considerations

• We are obliged to demand

=>

without theoretically inconsistency

• Put in dimensionless form

• Together with

or

• After substitution, we observe

• If we choose the constraint

• We see how the m’-equation is modified

thus

appears as first and second-order corrections for the two terms in the square bracket

Some constraints• Ghost-free conditions:

• Observational constraints on

Gravity Probe B for binary pulsar :

Strong magnetic field neutron star :

Ref.S. Arapoglu, C. Deliduman and K. Y. Eksi, ”Constraints on Perturbative f(R) Gravity via Neutron Stars”, JCAParXiv:1003.3179v3 [gr-qc]

Numerical Results

Profiles for poly-tropic EoS

• Our coupled ODEs are sensitive for small

• The physical solutions are fine-tuned for and

• In the following, we keep and adjust

For different • There exist solutions for and . • General feature: (1) The smaller the , the larger the . (2) The mass (radius) is smaller (bigger) for larger .

Profiles with • Ricci scalar and its derivative match the B.C. of the Schwarzschild vacuum solution.

• Ricci scalar deviates from .

• Mass function deviates much more from GR, whereas the does not. • The effective density matters.• Chandra limit of can be exceeded with .

For different with• Smaller can allow both larger mass and radius.• For ordinary matter, condition is required, therefore, we avoid for with at the center of NS.

Buchdahl Stability Bound

• In GR, we have but not (Buchdahl stability bound)

• Is there a corresponding relation for R2 model?

• In the R2 model with poly-tropic EoS, still holds for . (As we have seen from TABLE I.)

Summary• We have solved this model exactly rather than perturbatively.

• of the EoS is fine-tuned by the central values and hence the f(R) junction conditions.

• There can exist a EoS of with that has a mass exceeding the Chandra limit, i.e.

Thanks for your attention!!