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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
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Outline• Introduction
• The Coupled Ordinary Equations
• Junction & Boundary Conditions
• Numerical Results
• Buchdahl Stability Bound
• Summary
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Introduction
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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]
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Modified Gravity Action• The modified action
• After doing variation
with and
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The R2 Model• R2 model (Starobinsky model)
• Field equations:
• Trace equation:
=> Curvature relates to matter differentially rather than algebraically
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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?
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• We can approximate the density by
• For WD,
• For NS,
=>
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• 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.
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The Coupled Ordinary Equations
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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
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• For the sake of numerical set-up, we need to express
three coupled differential equations
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The coupled ODEs• After laborious calculations, the results are:
Modified TOV equation
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• The most different part from GR
=> Curvature relates to matter differentially rather than algebraically
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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:
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• 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
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Junction & Boundary Conditions
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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]
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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
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• 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.
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• 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
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What is the reasonable ?• Important observations on the dimensionless parameters of
our system:
• should be constrained by some multiplicative combinations of these parameters.
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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!!
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• Put in dimensionless form
• Second derivative of the Ricci scalar seems problematic
as !!
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• We can resolve it by the following considerations
• We are obliged to demand
=>
without theoretically inconsistency
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• Put in dimensionless form
• Together with
or
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• 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
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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]
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Numerical Results
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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
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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 .
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Profiles with • Ricci scalar and its derivative match the B.C. of the Schwarzschild vacuum solution.
• Ricci scalar deviates from .
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• Mass function deviates much more from GR, whereas the does not. • The effective density matters.• Chandra limit of can be exceeded with .
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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.
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Buchdahl Stability Bound
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• 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.)
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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.
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Thanks for your attention!!