Imprint of nuclear symmetry energy on gravitational waves from axial w-modes

Outline:1. Introduction of axial w-mode2. EOS constrained by recent terrestrial laboratory data3. Numerical Result and Discussion

collaborators Bao-An Li1 and Plamen Krastev2

Dehua Wen （文德华）

Department of Physics, South China Univ. of Tech.Department of Physics, Texas A&M University-Commerce

1Department of Physics and Astronomy, Texas A&M University-Commerce

2Department of Physics, San Diego State University

Please read Phys. Rev. C 80, 025801 (2009) for details

Axial mode: under the angular transformation θ→ π − θ, ϕ → π + ϕ, a spherical harmonic function with index ℓ transforms as (−1)ℓ+1 for the expanding metric functions.

The non-radial neutron star oscillations could be triggered by various mechanisms such as gravitational collapse, a pulsar “glitch” or a phase transition of matter in the inner core.

Polar mode: transforms as (−1)ℓ

Oscillating neutron star

1. Introduction of axial w-mode

Axial w-mode: not accompanied by any matter motions and only the perturbation of the space-time, exists for all relativistic stars, including neutron star and black holes.

One major characteristic of the axial w-mode is its high frequency accompanied by very rapid damping.

A network of large-scale ground-based laser-interferometer detectors (LIGO, VIRGO, GEO600, TAMA300) is on-line in detecting the gravitational waves (GW).

Theorists are presently try their best to think of various sources of GWs that may be observable once the new ultra-sensitive detectors operate at their optimum level.

MNRAS(2001)320,307

The importance for astrophysics

GWs from non-radial neutron star oscillations are considered as one of the most important sources.

Key equation of axial w-mode

Inner the star (l=2)

Outer the star

The equation for oscillation of the axial w-mode is give by1

dr

de

dr

d *

drerr

0*

where

or

]6)(6[ 33

2

mprrr

eV

][6

3

2

Mrr

eV

0)]([ 22

*

2

zrVdr

zd

ii 0

1 S.Chandrasekhar and V. Ferrari, Proc. R. Soc. London A, 432, 247(1991) Nobel prize in 1983

The standard axial w-mode is categorized as wI . The

high order axial w-modes are marked as the second w-

mode (wI2 -mode), the third mode (wI3 -mode) and so on.

An interesting additionally family of axial w-modes is ca

tegorized as wII .

2. EOS constrained by recent terrestrial laboratory data

Constrain by the flow data of relativistic heavy-ion reactions

P. Danielewicz, R. Lacey and W.G. Lynch, Science 298 (2002) 1592

1.M.B. Tsang, et al, Phys. Rev. Lett.

92, 062701 (2004)

2.  B. A. Li, L.W. Chen, and C.M. Ko,

Phys. Rep. 464, 113 (2008).

It was shown that only values of x in the range between −1 (MDIx-1) and 0 (MDIx0) are consistent with the isospin-diffusion and isoscaling data at sub-saturation densities.

Here we assume that the EOS can be extrapolated to supra-saturation densities according to the MDI predictions.

It was shown that only values of x in the range between −1 (MDIx-1) and 0 (MDIx0) are consistent with the isospin-diffusion and isoscaling data at sub-saturation densities.

Here we assume that the EOS can be extrapolated to supra-saturation densities according to the MDI predictions.1. L.W.Chen, C. M. Ko, and B. A. Li, Phys. Rev.

Lett. 94, 032701 (2005).

2.B. A. Li, L.W. Chen, and C.M. Ko,

Phys. Rep. 464, 113 (2008).

M-R

The w-modes are very important for astrophysical applications. The gravitational wave frequency of the axial w-mode depends on the neutron star’s structure and properties, which are determined by the EOS of neutron-rich stellar matter.

Heavy-ion reactions provide means to constrain the uncertain density behavior of the nuclear symmetry energy and thus the EOS of neutron-rich nuclear matter.

It is helpful to the detection of gravitational waves to investigate the imprint of the nuclear symmetry energy constrained by very recent terrestrial nuclear laboratory data on the gravitational waves from the axial w-mode.

Motivation

3. Numerical Result and Discussion

Frequency and damping time of axial w-mode

1

L. K. Tsui and P. T. Leung, MNRAS 357, 1029 (2005).

Scaling characteristic

The gravitational energy is calculated from1

1S.Weinberg, Gravitation and cosmology, (New York: Wiley,1972)

Eigen-frequency of wI2 scaled by the mass and the gravitational field energy

Tsui, et al, MNRAS, 357, 1029(2005)

Based on this linear dependence of the scaled frequency, the wII -mode is found to exist about compactness M/R>0.1078.

Exists linear fit

Conclusion 1. The density dependence of the nuclear symmetry energy affects significantly both the frequencies and the damping times of axial w-mode.

2. Obtain a better scaling characteristic, especially to th

e wI2-mode through scaling the eigen-frequency by the g

ravitational energy.

3. Give a general limit, M/R~0.1078, based on the linea

r scaling characteristic of wII, below this limit, wII-mode

will disappear. PHYSICAL REVIEW C 80, 025801 (2009)

Thanks

Appendix

The value of the isospin asymmetry δ at β equilibrium is determined by the chemical equilibrium and charge neutrality conditions, i.e., δ = 1 − 2xp with

Definition of radial and non-radial oscillation 1Rezzola, Gravitational waves from perturbed black holes and neutron stars

Radial oscillation:

Definition of radial and non-radial oscillation 2

Non-radial oscillation(Newton theory):

The main difference is the perturbation function.

Or in spherical harmonic function, the l=0 is radial oscillation, and l>=1 is non-radial oscillation.

PRD, 1999,60,104025

gravitational radiation from neutron star oscillations exhibits certain characteristic frequencies which are independent of the processes giving rise to these oscillations. These “quasi-normal” frequencies are directly connected to the parameters of neutron star (mass, charge and angular momentum) and are expected to be inside the bandwidth of the constructed gravitational wave detectors.

There are two classes of vector spherical harmonics (polar and axial) which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between the two families is their parity. Under the parity operator π (under the angular transformation θ→ π − θ, ϕ → π + ϕ), a spherical harmonic with index ℓ transforms as (−1)ℓ, the polar class of perturbations transform under parity in the same way, as (−1)ℓ, and the axial perturbations as (−1)ℓ+1. Finally, since we are dealing with spherically symmetric space-times the solution will be independent of m, thus this subscript can be omitted.

gravitational radiation from neutron star oscillations exhibits certain characteristic frequencies which are independent of the processes giving rise to these oscillations. These “quasi-normal” frequencies are directly connected to the parameters of neutron star (mass, charge and angular momentum) and are expected to be inside the bandwidth of the constructed gravitational wave detectors.

There are two classes of vector spherical harmonics (polar and axial) which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between the two families is their parity. Under the parity operator π (under the angular transformation θ→ π − θ, ϕ → π + ϕ), a spherical harmonic with index ℓ transforms as (−1)ℓ, the polar class of perturbations transform under parity in the same way, as (−1)ℓ, and the axial perturbations as (−1)ℓ+1. Finally, since we are dealing with spherically symmetric space-times the solution will be independent of m, thus this subscript can be omitted.

Definition of polar and axial

Definition of Axial oscillation 1

Thorne, APJ, 1967,149, 491

Definition of Axial oscillation 2

Note: in fact, now it is already found that there are gravitational waves in axial mode, that is, in the odd-parity pulsations.

Definition of polar oscillation

Polar oscillation:

Definition of Quasi-normal mode2

The meaning of the complex frequency

The eigen-frequencies of the modes are complex, the real part gives the pulsation rate and the imaginary part gives the damping time of the pulsation as a result of the emission of the gravitational radiation.

MNRAS(1992)255,119

Typical values of the frequencies and the damping times of various families of modes for a polytropic star (N = 1) with R = 8.86 km and M = 1.27M⊙ are given. p1 is the first p-mode, g1 is the first g-mode, w1 stands for the first curvature mode and wII for the slowest damped interface mode. For this stellar model there are no trapped modes11.

The frequency of different modes

Definition of w-modeTo the perturbation,

Only consider the space-time, that is, V = W = 0

So the perturbation equations for a specific multi-pole ℓ take the following simple form inside the star