Pulsars as a probe for the nuclear and quark matter equation of state

– p.1
Pulsars as a Probe for the Nuclear and Quark Matter
Equation of State
Jürgen Schaffner-Bielich
Institute for Theoretical Physics and
Heidelberg Graduate School for Fundamental Physics and
ExtreMe Matter Institute EMMI
AG 2011 Splinter Meeting on ’A fresh view of the radio sky: science
with LOFAR, SKA and its pathfinders’ Heidelberg, September 21, 2011

– p.2
Simon Weissenborn, Irina Sagert, Giuseppe Pagliara,
Matthias Hempel, JSB:
Quark matter in massive neutron stars,
e-Print: arXiv:1102.2869 [astro-ph.HE], ApJL in press

EoS and mass-radius relation of compact stars
– p.3
(Weber, Negreiros, Rosenfield, Steijner 2006)
many, many different equation of state EoS . . .
result in various different mass-radius relation for compact stars
Tolman-Oppenheimer-Volkoff (TOV) equation: one-to-one relation between
EoS and mass-radius relation

Constraints on the Mass–Radius Relation (Lattimer and Prakash 2004)
Mass (solar)
2.5
2.0
1.5
1.0
GR
P <
causality
AP3
ENG
AP4
J1614-2230
SQM3
J1903+0327 FSU
SQM1
J1909-3744
PAL6 GM3
GS1
Double NS Systems
MPA1
MS1
PAL1
MS2
MS0
0.5
rotation
Nucleons Nucleons+Exotic Strange Quark Matter
0.0
7 8 9 10 11 12 13 14 15
Radius (km)
spin rate from PSR B1937+21 of 641 Hz: R < 15.5 km for M = 1.4M ⊙
Schwarzschild limit (GR): R > 2GM = R s
causality limit for EoS: R > 3GM
mass limit from PSR J1614-2230 (red band): M = (1.97±0.04)M ⊙
– p.4

Nuclear Equation of State as Input in Astrophysics
supernovae simulations: T = 1–50 MeV, n = 10 −10 –2n 0
proto-neutron star: T = 1–50 MeV, n = 10 −3 –10n 0
global properties of neutron stars: T = 0, n = 10 −3 –10n 0
neutron star mergers: T = 0–100 MeV, n = 10 −10 –10n 0
– p.5

Phase Diagram of Quantum Chromodynamics QCD
– p.6
T
T c
early universe
RHIC, LHC
Quark−Gluon
Plasma
FAIR
Hadrons
nuclei
Color
Superconductivity
neutron stars
m / 3
Early universe at zero density and high temperature
N
µ
c
µ
neutron star matter at small temperature and high density
first order phase transition at high density (not deconfinement!)
probed by heavy-ion collisions at GSI, Darmstadt (FAIR)

– p.7
Structure of a Neutron Star (Fridolin Weber)
quark−hybrid
star
N+e
N+e+n
traditional neutron star
n,p,e, µ
n superfluid
hyperon
star
s u
p
e
r
c
o
n d
u
c
t
i
n
g
p
neutron star with
pion condensate
absolutely stable
strange quark
matter
µ
u d s
Σ,Λ,Ξ,∆
u,d,s
quarks
H
n,p,e, µ
K −
π −
r o
t
n o s
crust
Fe
10 6 g/cm 3
10 11 g/cm 3
10 14 g/cm 3
m s
strange star
R ~ 10 km
M ~ 1.4 M
nucleon star

Maximum masses of neutron stars with hyperons
– p.8
Brueckner-Hartree-Fock
calculation with most recent soft
core Nijmegen potential ESC08
includes repulsive three-body
forces (TBF, UIX’)
overall findings: M < 1.4M ⊙
when hyperons are included
higher masses possible within
RMF model and SU(3)
symmetry (Weissenborn,
Chatterjee, JSB, in preparation)
(Schulze and Rijken 2011)
other possible solution:
quark matter core
a stiff

Selfbound Star versus Ordinary Neutron Star
– p.9
(Hartle, Sawyer, Scalapino (1975!))
selfbound stars:
vanishing pressure at a finite
energy density
mass-radius relation starts at the
origin (ignoring a possible crust)
arbitrarily small masses and radii
possible
neutron stars:
bound by gravity, finite pressure for
all energy density
mass-radius relation starts at large
radii
minimum neutron star mass:
M ∼ 0.1M ⊙ with R ∼ 200 km

Quark Star Masses: Unpaired Case
– p.10
Use free gas of quarks with a term from interactions and from a
vacuum energy:
Ω QM = ∑
i=u,d,s,e
Ω i + 3µ4
4π 2(1−a 4)+B eff
Effective model with an expansion in the chemical potential µ
Two parameters: effective bag constant B eff and interaction
parameter a 4
2-flavour constraint: nuclei do not collapse to (u,d) quark matter!
3-flavour constraint: strange (u,d,s) quark matter shall be more
stable than nuclear matter, so that selfbound quark stars dubbed
strange stars can exist

Quark Star Masses: Unpaired Case
– p.11
(Weissenborn, Sagert, Pagliara, Hempel, JSB 2011)
Kepler line: mass shedding limit for 716 Hz
(highest observed pulsar frequency)
green region: allowed parameter space from maximum pulsar mass
corrections from interactions are needed (a 4 < 1) to be compatible with
observations!

Quark Star Masses: effects of quark pairing
– p.12
Add to a free gas of quarks terms from interaction, from pairing and
from an vacuum energy:
Ω CFL = 6 π 2 ∫ ν
0
dp p 2 (p−µ)+ 3 π 2 ∫ ν
+(1−a 4 ) 3µ4
4π − 3∆2 µ 2
2 π 2
where ν = 2µ− √ µ 2 −m 2 s/3.
0
+B eff
dp p 2 ( √ p 2 +m 2 s −µ)
∆: gap energy of the color-superconducting phase
(normally ∆ ≤ 100 MeV)
fix strange quark mass to m s = 100 MeV
set for simplicity a 4 = 0

Quark Star Masses: effects of quark pairing
(Weissenborn, Sagert, Pagliara, Hempel, JSB 2011)
two constraints on quark matter: 2-flavour and 3-flavour line
green region: allowed parameter space from maximum pulsar mass
a gap of at least ∆ = 20 MeV is needed to be compatible with observations
pulsar masses above 1.9M ⊙ start to constrain QCD parameters!
additional interactions needed for pulsar masses well above 2.3M ⊙
– p.13

Hybrid Stars with a stiff nuclear EoS
– p.14
2.8
2.6
a 4 =0.4
Maximum Mass [M solar ]
2.4
2.2
2
1.8
1.6
1.4
a 4 =0.5
120 140 160 180 200 220 240 260 280 300
B eff
1/4 [MeV]
a 4 =1.0
PSR J1614-2230
Maxwell hybrid stars
1.67 M
Maxwell, quark core
solar
Gibbs, quark core
Gibbs, mixed core
n c ~ 0.1 1/fm 3
1.44 M
n c ~ 0.2 1/fm 3
solar n c ~ 0.3 1/fm 3
n c ~ 0.4 1/fm 3
n c ~ 0.5 1/fm 3
(Weissenborn, Sagert, Pagliara, Hempel, JSB 2011)
nuclear phase: relativistic mean field model with parameter set NL3 (fitted to
properties of nuclei)
match with Gibbs (lines) or Maxwell construction (shaded area)
solid lines: pure quark matter cores, dashed lines: mixed phase cores

Hybrid Stars with a soft nuclear EoS
2.3
2.2
a 4 =0.4
Maximum Mass [M solar ]
2.1
2
1.9
1.8
a 4 =0.5
a 4 =1.0
B eff
1/4 [MeV]
PSR J1614-2230
1.7 1.67 M solar
Maxwell, hybrid stars
Maxwell, quark core
1.6
Gibbs, quark core
Gibbs, mixed core
n
1.5
c ~ 0.1 1/fm 3
1.44 M n solar c ~ 0.2 1/fm 3
n c ~ 0.3 1/fm 3
1.4
n c ~ 0.4 1/fm 3
n c ~ 0.5 1/fm 3
n c ~ 0.6 1/fm 3
1.3
120 140 160 180 200 220 240 260 280 300
(Weissenborn, Sagert, Pagliara, Hempel, JSB 2011)
nuclear phase: relativistic mean field model with parameter set TM1 (fitted to
properties of nuclei)
match with Gibbs (lines) or Maxwell construction (shaded area)
solid lines: pure quark matter cores, dashed lines: mixed phase cores
no pure quark cores compatible with data for a soft nuclear EoS
– p.15

Hybrid Stars with a NJL model
– p.16
(Bonanno and Sedrakian 2011)
uses Nambu-Jona-Lasinio model for quark matter
matches to nuclear EoS with hyperons (RMF with set NL3)
2SC quark matter below green line
δ = R CFL /R: amount of CFL quark matter

Maximum central density of a compact stars
– p.17
(Lattimer and Prakash 2011)
maximally compact EoS: p = s(ǫ−ǫ c ) with s = 1
stiffest possible EoS (Zeldovich 1961)
gives upper limit on compact star mass: M max = 4.2M ⊙ (ǫ sat. /ǫ f ) 1/2
(Rhoades and Ruffini 1974, Hartle 1978, Kalogera and Baym 1996, Akmal, Pandharipande,
Ravenhall 1998)

Maximum Masses of Neutron Stars – Causality
3.5
3
Skyrme Bsk8
n crit =2.5 n 0
n crit =2 n 0
n crit =3 n 0
2.5
M [M solar ]
2
1.5
1
0.5
9 10 11 12 13 14 15 16
R [km]
(Sagert, Sturm, Chatterjee, Tolos, JSB in preparation)
Skyrme parameter set BSK8: fitted to masses of all known nuclei
above a fiducial density (determined from the analysis of the KaoS heavy-ion
data) transition to stiffest possible EoS
causality argument: p = ǫ−ǫ c above the fiducial density ǫ f
Rhoades, Ruffini (1974), Kalogera, Baym (1996): M max = 4.2M ⊙ (ǫ 0 /ǫ f ) 1/2
=⇒ new upper mass limit of about 2.8M ⊙ from heavy-ion data!
– p.18

Maximum Masses of Neutron Stars – Causality
3.5
3
Skyrme SLy4
n crit =2.5 n 0
n crit =2 n 0
n crit =3 n 0
2.5
M [M solar ]
2
1.5
1
0.5
9 10 11 12 13 14 15 16
R [km]
(Sagert, Sturm, Chatterjee, Tolos, JSB in preparation)
Skyrme parameter set Sly4: fitted to properties of spherical nuclei
above a fiducial density (determined from the analysis of the KaoS heavy-ion
data) transition to stiffest possible EoS
causality argument: p = ǫ−ǫ c above the fiducial density ǫ f
Rhoades, Ruffini (1974), Kalogera, Baym (1996): M max = 4.2M ⊙ (ǫ 0 /ǫ f ) 1/2
=⇒ new upper mass limit of about 2.8M ⊙ from heavy-ion data!
– p.18

Maximum Masses of Neutron Stars – Causality
3.5
3
RMF TM1
n crit =2.5 n 0
n crit =2 n 0
n crit =3 n 0
2.5
M [M solar ]
2
1.5
1
0.5
9 10 11 12 13 14 15 16
R [km]
(Sagert, Sturm, Chatterjee, Tolos, JSB in preparation)
RMF parameter set TM1: fitted to properties of spherical nuclei
above a fiducial density (determined from the analysis of the KaoS heavy-ion
data) transition to stiffest possible EoS
causality argument: p = ǫ−ǫ c above the fiducial density ǫ f
Rhoades, Ruffini (1974), Kalogera, Baym (1996): M max = 4.2M ⊙ (ǫ 0 /ǫ f ) 1/2
=⇒ new upper mass limit of about 2.8M ⊙ from heavy-ion data!
– p.18

– p.19
Third Family of Compact Stars (Gerlach 1968)
Ethirdfamily neutronstars whitedwarfs
D C A
F G H I
instablemodes
R B
M=M
(Glendenning, Kettner 2000; Schertler, Greiner, JSB, Thoma 2000)
third solution to the TOV equations besides white dwarfs and neutron stars,
solution is stable!
generates stars more compact than neutron stars!
possible for any first order phase transition!

Quark star twins? (Fraga, JSB, Pisarski (2001))
2
weak transition
Λ=3µ
1.5
n c
=3n 0
2.5n 0
M/M sun
1
0.5
strong transition
Λ=2µ
new
stable
branch
2n 0
0
hadronic EoS
0 5 10 15
Radius (km)
Weak transition: ordinary neutron star with quark core (hybrid star)
Strong transition: third class of compact stars possible with maximum masses
M ∼ 1M ⊙ and radii R ∼ 6 km
Quark phase dominates (n ∼ 15n 0 at the center), small hadronic mantle
– p.20

Summary
– p.21
equation of state (EoS) determines the maximum
mass and its radius
new hadronic degrees of freedoms normally soften
the EoS
but quark matter can also stiffen the EoS
strong QCD phase transition leads to a third family of
compact stars
pulsar mass measurements can severely constrain
models of QCD
unique opportunity to test QCD at extreme conditions

Onset of Hyperons in Neutron Star Matter
Hyperons appear at n ≈ 2n 0 ! (based on hypernuclear data!)
relativistic mean–field models
(Glendenning 1985; Knorren, Prakash, Ellis 1996; JS and Mishustin 1996)
nonrelativistic potential model (Balberg and Gal 1997)
quark-meson coupling model (Pal et al. 1999)
relativistic Hartree–Fock (Huber, Weber, Weigel, Schaab 1998)
Brueckner–Hartree–Fock
(Baldo, Burgio, Schulze 1998, 2000; Vidana et al. 2000; Schulze, Polls, Ramos, Vidana 2006)
chiral effective Lagrangian using SU(3) symmetry
(Hanauske et al. 2000; Schramm and Zschiesche 2003; Dexheimer and Schramm 2008)
density-dependent hadron field theory (Hofmann, Keil, Lenske 2001)
G-matrix calculation (Nishizaki, Takatsuka, Yamamoto 2002)
RG approach with V low k (Djapo, Schäfer, Wambach 2008)
⇒ neutron stars are giant hypernuclei !!!
– p.22

Impact of hyperons on the maximum mass
– p.23
neutron star with nucleons
and leptons only:
M ≈ 2.3M ⊙
substantial decrease of the
maximum mass due to
hyperons!
maximum mass for “giant
hypernuclei”: M ≈ 1.7M ⊙
(Glendenning and Moszkowski 1991)
noninteracting hyperons
result in a too low mass:
M < 1.4M ⊙ !

Maximum mass and modern many-body approaches
– p.24
modern many-body calculations
(using Nijmegen soft-core YN potential)
Vidana et al. (2000): M max = 1.47M ⊙ (NN and YN interactions),
M max = 1.34M ⊙ (NN, NY, YY interactions)
Baldo et al. (2000): M max = 1.26M ⊙
(including three-body nucleon interaction)
Schulze et al. (2006): M max < 1.4M ⊙
Djapo et al. (2008): M max < 1.4M ⊙
Schulze and Rijken (2011): M max < 1.4M ⊙ (ESC08)
too soft EoS, too low masses!
resolution: make EoS stiffer by the presence of quark matter!?!

Phases in Quark Matter (Rüster et al. (2005))
60
50
40
NQ
2SC
uSC
T [MeV]
30
guSC→
20
χSB
g2SC
10
CFL
NQ
←gCFL
0
320 340 360 380 400 420 440 460 480 500
µ [MeV]
first order phase transition based on symmetry arguments!
phases of color superconducting quark matter in β equilibrium:
normal (unpaired) quark matter (NQ)
two-flavor color superconducting phase (2SC), gapless 2SC phase
color-flavor locked phase (CFL), gapless CFL phase, metallic CFL phase
(Alford, Rajagopal, Wilczek, Reddy, Buballa, Blaschke, Shovkovy, Drago, Rüster, Rischke,
Aguilera, Banik, Bandyopadhyay, Pagliara, . . . )
– p.25

X-Ray burster EXO 0748–676 and Quark Matter
Mass M [M ]
2.4
2.0
1.6
1.2
0.8
Hybrid, mixed
(CFL+APR)
Quark star
(QCD O(α s 2 ))
Neutron star
(DBHF)
2σ
EXO 0748-676
Hybrid
(2SC+DBHF)
Hybrid
(2SC+HHJ)
0.4
7 8 9 10 11 12 13 14 15
Radius R [km]
analysis of Özel (Nature 2006): M ≥ 2.10±0.28M ⊙ and R ≥ 13.8±1.8 km,
claims: ’unconfined quarks do not exist at the center of neutron stars’!
reply by Alford, Blaschke, Drago, Klähn, Pagliara, JSB (Nature 445, E7
(2007)): limits rule out soft equations of state, not quark stars or hybrid stars!
multiwavelength analysis of Pearson et al. (2006): data more consistent with
M = 1.35M ⊙ than with M = 2.1M ⊙
– p.26
1σ

Signals for a Third Family/Phase Transition?
– p.27
mass-radius relation: rising twins (Schertler et al.,
2000)
spontaneous spin-up of pulsars (Glendenning, Pei,
Weber, 1997)
delayed collapse of a proto-neutron star to a black
hole (Thorsson, Prakash, Lattimer, 1994)
bimodal distribution of pulsar kick velocities (Bombaci
and Popov, 2004)
collapse of a neutron star to the third family?
(gravitational waves, γ-rays, neutrinos)
secondary shock wave in supernova explosions?
gravitational waves from colliding neutron stars?

Signals for Strange Stars?
– p.28
similar masses and radii, cooling, surface (crust), . . . but
look for
extremely small mass, small radius stars (includes
strangelets)
strange dwarfs: small and light white dwarfs with a
strange star core (Glendenning, Kettner, Weber, 1995)
super-Eddington luminosity from bare, hot strange
stars (Page and Usov, 2002)
conversion of neutron stars to strange stars (explosive
events!)
. . .