Metastability of Hadronic Compact Stars

I. Vidaña

& I. Bombaci, P. K. Panda, C. Providência

“The Complex Physics of Compact Stars” Ladek Zdroj, Poland, 24-29 February 2008

arXiv:0802.1794; PRD in press

In this work:

We perform a systematic study of the metastability of pure hadronic compact stars with respect to the conversion to quark stars using different relativistic models for the hadronic EoS: Non Linear Walecka Model (NLWM) & Quark Meson Coupling (QMC).

We explore the effect of different hyperon couplings on the critical mass and on the stellar conversion energy, finding that the increase of the hyperon coupling shift the bulk transition point for quark deconfinement to higher densities and makes the conversion to quark stars less likely.

For the QMC model, the formation of a quark star is only possible with a soft quark matter EoS.

Both QMC and GM1 with the largest hyperon-meson couplings predict critical masses which may be as high as 1.9-2.1 M, compatible with highly massive compact stars, suchs as the of the millisecond pulsar PSR B1516+02B and nearly the one PSR J1748-2021B.

Astrophysical Scenario

Nucleation of quark matter in neutron stars has been studied by many authors, due to its potential connection with explosive events such as supernovae and gamma-ray bursts.

Thermal nucleation: (Horvath et al.,1994; Olesen & Madsen 2004; Benvenuto & Lugones 1999)

The prompt formation of a critical-size drop of QM via thermal activation is possible for T > 2-3 MeV Pure hadronic stars are converted to quark stars within the first seconds after their birth.

Quantum nucleation:(Grassi 1998; Iida & Sato 1998, Berezhiani et al., 2003, Bombaci et al., 2004, Drago et al., 2004)

It is possible that the star survives the early stages of its evolution asa pure hadronic star. In this case, nucleation of QM would be triggeredby quantum fluctuations in degenerate (T=0) neutrino-free hadronicmatter.

However, neutrino trapping in the protoneutron star phase strongly precludes the formation of a quark matter phase

Formation of a quark matter bubble at the centre of a Neutron Star (I)

Hadron matterHadron matter -stable -stable quark-matter bubble

Direct nucleation of the -stable quark matter: high order weak process suppressed by a factor ~ GF

2N/3, with N=100-1000.

Ruled out: even when the final state has a lower energy

Berezhiani et al. 2003 (unpaired)Drago, Lavagno & Pagliara 2004 (CFL)

Formation of a quark matter bubble at the centre of a Neutron Star (II)

W ~ 10-8 sS ~ 10-23 s

Hadron matterHadron matter -stable-stable quark-matter bubble

Non-Non--stable-stable quark-matter bubble

Q* (Non- stable)

Q* (Non- stable)

Has the intermediate phase lower energy than hadron matter ?

The intermediate non -stable quark phase

Each flavor is color neutral

Flavor is conserved

Iida & Sato 1998 ; Bombaci, Parenti & Vidaña 2004

Lifshitz-Kagan quantum nucleation theoryQuantum fluctuation of a virtual drop of QM in HM

( ) ( ) ( )RURM+dtdRRM=L −⎟

⎠⎞⎜

⎝⎛−−

2

1

€

M R( ) = 4πρ H 1−nQ*

nH

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

R3

€

U R( ) = 43

πnQ* μQ* − μ H( )R3 +4πσR2 + 8πγR + E c

€

σ =10 − 50MeV / fm2

Nucleation Time

Action over andunder the barrier

Nucleation time

€

ν 0 = dIdE ⎛ ⎝ ⎜

⎞ ⎠ ⎟−1

;E = Eo

( )⎟⎠⎞⎜⎝

⎛− h0

0 exp EA=p

€

I E( ) = 2 dR0

R1

∫ 2M R( ) + E −U R( )[ ] E −U R( )[ ]

€

A E( ) = 2 dRR1

R2

∫ 2M R( )+ E −U R( )[ ] U R( ) − E[ ]

€

= ν 0 p0Nc( )−1;Nc ≈1048

Oscillation frequency of the virtual drop inside the potential well and Penetrabilityof the potential barrier (WKB)

Critical mass of metastable hadronic stars

Definition: Mcr = MHS( = 1 yr)

Hadronic stars with MHS< Mcr are metastable with = 1 yr to infinity

Hadronic stars with MHS> Mcr are very unlikely observed

“The critical mass Mcr plays the role of an effective maximum mass for the hadronic branch of compact stars”

Berezhiani et al. 2003 ; Bombaci et al. 2004

Few words on the EoS considered …

Non Linear Walecka ModelWe use the Glendenning-Moszkowski (GM) parametrizations GM1 & GM3 of the NLWM (PRL 67, 2414 (1991)), where the hyperon-nucleon couplings, xσgYσ/gNσ xgY/gN and xgY/gN , are constrained by the binding of the hyperon in nuclear matter

€

BΛ

A ⎛ ⎝ ⎜

⎞ ⎠ ⎟= −28MeV = xωgωω0 − xσ gσ σ

Neutron star masses, in addition, restrict xσ to the range 0.6 - 0.8. Here, we will take x xσ and will consider xσ =0.6, 0.7, 0.8.

Quark Meson Coupling ModelBaryons described as a system of non-overlaping spherical bags containing three valence quarks interacting by the exchange of σ, and mesons coupled directly to the confined quarks. (PLB 200, 235 (1988)). Here we will take x 0.7, x=0.78 and xσ is an output of the model ~ 0.7.

Quark phase: MIT bag model (Farhi & Jaffe, PRD 30, 2379 (1984))

Hadronic Equation of State

Higher value of the hyperon couplings stiffer EoS.

Onset of hyperons at higher densities for larger values of the couplings.

QMC EoS softer/stiffer than NLWM.

Gibbs free energy & bulk transition point for quark deconfinement

The lower the values of hyperon couplings, the softer the EoS and the lower the pressure P0 at the crossing between the hadronic and the Q* phase lower critical masses for smaller hyperon coupling values.

NLWM (GM1) QMC

Stable, mestastable and unstable hadron star configurations (I)

Very narrow metastability region for QMC. The formation of a quark star is only possible in this model for a soft quark matter EoS (i.e., for small values of B).

σ=30 MeV/fm2

Stable, mestastable and unstable hadron star configurations (II)

NLWM (GM1) QMC

When Mcr star is almost on the top of P0, these stars lie on or close to the plateau that contains the maximum mass configuration.

A large separation between these two configurations corresponds to a phase transition which occurs during the rise of the MP curve before the plateau.

NLW

M (G

M1)

Q

MC

Hyp

eron

cou

plin

g

Crit

ical

Mas

sCrit

ical

Mas

s

Bag

cos

natn

t

Large critical masses due to the softness of the QMC EoS

Compatible with highly-massive compact stars, such as the one associated to the millisecond pulsar PSR B1516+02B (1.94(+0.17-0.19) M (1σ)) , and nearly the one PSR J1748-2021B (2.74 (+0.41-0.51) M (2σ))

M=2.081 M, R=12.6 km, fY,cr ~ 30%, RY ~ 8.7 km.

Summary & Conclusions 4564)

We have performed a systematic study of the metastability of pure hadronic compact stars with respect to the conversion to quark stars using different relativistic models for the EoS: Non Linear Walecka Model (NLWM) & Quark Meson Coupling (QMC).

We have explored the effect of different hyperon couplings on the critical mass and on the stellar conversion energy, finding that the increase of the hyperon coupling shift the bulk transition point for quark deconfinement to higher densities and makes the conversion to quark stars less likely.

For the QMC model, the metastability region is very narrow. The EoS is very sofy and therefore the onset of hyperons occurs at quite high densities, which gives rise to large critical masses. The converstion to a quark star will occur only for a small value of the bag constant.

Both QMC and GM1 with the largest hyperon-meson couplings predict critical masses which may be as high as 1.9-2.1 M, compatible with the masses of the millisecond pulsar PSR B1516+02B and nearly the one PSR J1748-2021B

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