Bulk viscosity of baryonic matter with trapped neutrinos ... · Bulk viscosity of baryonic matter with trapped neutrinos and its application to neutron star mergers Arus Harutyunyan

Bulk viscosity of baryonic matter with trappedneutrinos and its application to neutron star mergers

Arus Harutyunyan

Collaborators: A. Sedrakian, M. AlfordBased on: arXiv:1907.04192

Justus-Liebig-Universitat Gießen, October 23, 2019

October 23, 2019

Outline

Introduction & motivationUrca processes and bulk viscosityDensity functional theoryResults for bulk viscosityConclusions

Introduction & motivation

Compact stars


The structure of a neutron star


Compact-star binaries

Compact stars are natural laboratories which allow us to study the properties of nuclearmatter under extreme physical conditions (strong gravity, strong magnetic fields, etc.).

The recent detection of gravitational and electromagnetic waves originating from blackhole or neutron star mergers motivates studies of compact binary systems.

Such studies might place constraints on the properties of compact star parameters andcontain useful information about the properties of extremely hot and dense matter.

Various physical processes in the compact binary systems can be modelled in theframework of general-relativistic hydrodynamics simulations.

Transport coefficients are key inputs in hydrodynamic modelling of compact star mergersas they measure the energy dissipation rate in hydrodynamic evolution of matter.

The bulk viscosity might affect the hydrodynamic evolution of mergers by damping thedensity oscillations which can affect the form of the gravitational signal.

Our aim is to study the bulk viscosity in dense baryonic matter for temperatures relevantto neutron star mergers and supernovas T ≥ 5 MeV.

At these temperatures neutrinos are trapped in matter, and the bulk viscosity arises fromweak interaction (neutron decay and electron capture) processes.


Relativistic hydrodynamics and bulk viscosity

Hydrodynamic evolution of a relativistic system is described by means of theenergy-momentum tensor and particle current which obey the conservation laws

∂µTµν = 0, ∂µNµ = 0.

For ideal hydrodynamic, i.e., without dissipation

Tµν0 = ε0uµuν − p0∆µν , Nµ0 = n0uµ,

and the system of conservation laws is closed by an equation of state p0 = peq(ε0, n0).

For a dissipative fluid with velocity gradients

Tµν = εuµuν − p∆µν + πµν , Nµ = nuµ.

In this case the pressure obtains a non-equilibrium contribution

p = peq + Π.

where the bulk viscous pressure reads Π = −ζθ with θ = ∂µuµ.

Thus, the bulk viscosity ζ describes dissipation in the case where pressure falls out ofequilibrium on uniform expansion/contraction.


Literature on bulk viscosity

R. F. Sawyer, Damping of neutron star pulsations by weak interaction processes,Astrophys. J. 237 (1980) 187-197.

R. F. Sawyer, Bulk viscosity of hot neutron-star matter and the maximum rotation rates ofneutron stars, Phys. Rev. D 39 (1989) 3804-3806.

P. Haensel and R. Schaeffer, Bulk viscosity of hot-neutron-star matter from direct URCAprocesses, Phys. Rev. D 45 (1992) 4708-4712.

P. B. Jones, Bulk viscosity of neutron-star matter, Phys. Rev. D 64 (2001) 084003.

M. G. Alford, S. Mahmoodifar and K. Schwenzer, Large amplitude behavior of the bulkviscosity of dense matter, Journal of Physics G Nuclear Physics 37 (2010) 125202,[1005.3769].

M. G. Alford, L. Bovard, M. Hanauske, L. Rezzolla and K. Schwenzer, ViscousDissipation and Heat Conduction in Binary Neutron-Star Mergers, Physical ReviewLetters 120 (2018) 041101, [1707.09475].

D. G. Yakovlev, M. E. Gusakov and P. Haensel, Bulk viscosity in a neutron star mantle,Mon. Not. RAS 481 (2018) 4924-4930, [1809.08609].

A. Schmitt and P. Shternin, Reaction rates and transport in neutron stars, The Physicsand Astrophysics of Neutron Stars 457 (2018) 455-574 [1711.06520].

M. Alford and S. Harris, Damping of density oscillations in neutrino-transparent nuclearmatter, Phys. Rev. C 100 (2019) 035803, [1907.03795].

Urca processes and bulk viscosity

Urca process rates

We consider a simple composition of baryonic matter consisting of neutrons, protons,electrons and neutrinos. The simplest weak-interaction processes are the following(direct) Urca processes

n p + e− + νe (neutron decay process) (1)

p + e− n + νe (electron capture process) (2)

In β-equilibrium the chemical potentials of particles obey the relationµn + µν = µp + µe. Out of β-equilibrium in general implies an imbalance

µ∆ ≡ µn + µν − µp − µe 6= 0.

as a measure of deviation from β-equilibrium. The rate at which µ∆ relaxes to zero is ameasure of speed at which the constitution of matter adjusts to a change in pressure.The β-equilibration rate for the neutron decay is given by

Γ1p(µ∆) =

∫dΩ∑

si

|MUrca|2 f (p′) f (k′) f (k) f (p) (2π)4 δ(4)(p + k + k′ − p′),

with f (p) = 1− f (p). Similar expressions can be written also for Γ1n, Γ2p and Γ2n.The squared matrix element of Urca processes is

[G2 = G2

F cos2 θc(1 + 3g2A)]]∑

si

|MUrca|2 = 32G2(k · p′)(p · k′) ' 32G2p0p′0k0k′0.


Density oscillations in neutron-star matter

Consider now small-amplitude density oscillations in baryonic matter with frequency ω

nB(t) = nB0 + δnB(t), nL(t) = nL0 + δnL(t), δnB(t), δnL(t) ∼ eiωt.

The baryon and lepton number conservation ∂ni/∂t + div (niv) = 0 implies

δni(t) = −θ

iωni0, i = B, L, θ = div v.

The oscillations cause perturbations in particle densities nj(t) = nj0 + δnj(t), due towhich the chemical equilibrium of matter is disturbed leading to a small shiftµ∆ = δµn + δµν − δµp − δµe, which can be written as

µ∆ = (Ann − Apn)δnn + Aννδnν − (App − Anp)δnp − Aeeδne, Aij =

(∂µi

∂nj

)0.

The off-diagonal elements Anp and Apn are non-zero because of the cross-species stronginteraction between neutrons and protons.

If the weak processes are turned off, then a perturbation conserves all particle numbers

∂

∂tδnj(t) + θnj0 = 0, δnj(t) = −

θ

iωnj0.


Chemical balance equations and bulk viscosity

Out of equilibrium the chemical equilibration rate to linear order in µ∆ is given by

Γp − Γn = λµ∆, λ > 0.

The rate equations which take into account the loss and gain of particles read as

∂

∂tδnn(t) + θnn0 = −λµ∆(t),

∂

∂tδnp(t) + θnp0 = λµ∆(t).

Solving these equations we can compute the pressure out of equilibrium

p = p(nj) = p(nj0 + δnj) = p0 + δp = peq + δp′,

where the non-equilibrium part of the pressure - the bulk viscous pressure, is given by

Π ≡ δp′ =∑

j

(∂p∂nj

)0δn′j =

∑ij

ni0Aijδn′j .

The bulk viscosity is then identifined from Π = −ζθ

ζ =C2

AλA

ω2 + λ2A2

with susceptibilities A = −1

nB

(∂µ∆

∂xp

)nB

and C = nB

(∂µ∆

∂nB

)xp

.


Limiting cases for bulk viscosity

In the low-temperature limit the matter is ν-transparent, and the equilibration rate λ reads

λtrans =17

240πm∗2G2T4pFeθ(pFp + pFe − pFn),

where pFi are Fermi-momenta of particles. The θ-function blocks direct Urca processesat densities where pFp + pFe < pFn ⇒ modified Urca processes should be included.

The neutrino-trapped β-equilibration rate for strongly degenerate matter is given by

λtrap =1

12π3m∗2G2T2pFepFν(pFe + pFν − |pFn − pFp|).

In contrast to the ν-transparent case here the rate is finite in the low-temperature limit⇒ modified Urca reactions are not required to be included.

Because in the degenerate regime pFi T , the equilibration rate is alwals much larger inthe case of trapped neutrinos.

In the high-frequency (ω λA), low-frequency (ω λA) limits and at the maximumof the bulk viscosity (ω = λA) we find

ζhigh =C2λ

ω2, ζlow =

C2

λA2, ζmax =

C2

2Aω.

Density functional theory

Beta-equilibrated nuclear matter

We use the density functional theory approach to the nuclear matter, which is based onphenomenological baryon-meson Lagrangians of the type proposed by Walecka and others.

The Lagrangian density of matter is given by

L =∑

N

ψN

[γµ(

i∂µ − gωNωµ −12

gρNτρµ

)− m∗N

]ψN +

∑λ

ψλ(iγµ∂µ − mλ)ψλ

+12∂µσ∂µσ −

12

m2σσ

2 − U(σ)−14ωµνωµν +

12

m2ωω

µωµ −14ρµνρµν +

12

m2ρρµρµ.

The pressure of baryonic matter is given by

PN = −m2σ

2σ2 − U(σ) +

∑λ

gλ6π2

∫ ∞0

k4 dk(k2 + m2

λ)1/2

[f (Eλk − µλ) + f (Eλk + µλ)

]+

m2ω

2ω2

0 +m2ρ

2ρ2

03 +∑

N

2JN + 16π2

∫ ∞0

k4 dk(k2 + m∗2

N )1/2

[f (EN

k − µ∗N) + f (EN

k + µ∗N)]

Here m∗N = mN − gσNσ and µ∗N = µN − gωNω0 − gρNρ03I3 are the nucleon effective mass

and effective chemical potentials, respectively; I3 is the third component of nucleon isospin and

σ, ω0 and ρ03 are the mean values of the meson fields.


Meson mean-fields and susceptibilities

The mean-field values of the ω and ρ mesons are given by

gωω0 =

(gωmω

)2

(nn + np), gρρ03 =12

(gρmρ

)2

(np − nn).

The mean value of the scalar field is given by

gσσ = −gσm2σ

∂U(σ)

∂σ+

1π2

(gσmσ

)2∑N

∫ ∞0

p2dpm∗√

p2 + m∗2[fN(p) + fN(p)].

The “beta-disequilibrium–proton-fraction” susceptibility in low-T limit is given by

A =π2

m∗

(1

pFn+

1pFn

)+π2

p2Fe

+2π2

p2Fν

+

(gρmρ

)2

.

The “beta-disequilibrium–baryon-density” susceptibility in low-T limit reads

C =p2

Fn − p2Fp

3m∗+

pFν − pFe

3+

nn − np

2

(gρmρ

)2

+ nBp2

Fn − p2Fp

2m∗2

(gσmσ

)2

.


Particle fractions in equilibrium

The particle fractions are found from β-equilibrium conditions µn + µν = µp + µe andµµ = µe, the charge neutrality condition np = ne + nµ, the baryon number conservationnB = nn + np, and the lepton number conservation nl + nνl = nL = YLnB.

1 2 3n

B/n

0

νe

νµ

νe

1 2 3n

B/n

0

10-2

10-1

100

Yi n

p

e−

µ−

T = 5 MeV T = 50 MeV

YL = 0.1

(a) (b)

1 2 3n

B/n

0

1 2 3n

B/n

0

10-2

10-1

100

Yi

np, e

−

νe

µ−

T = 5 MeV T = 50 MeV

YL = 0.4

(a) (b)

We consider two cases: (i) YL = 0.1 for both flavors, typical for neutron star mergers;(ii) YLe = 0.4 and YLµ = 0 typical for matter in supernovae and proto-neutron stars.

The particle fractions are not sensitive to the temperature for the given value of YL.

In the low-density and high-temperature regime the net neutrino density becomesnegative, indicating that there are more anti-neutrinos than neutrinos in that regime.

Merger matter has much smaller electron neutrino fraction than supernova matter.


The equation of state for two models of nuclear matter

1 2 3n

B/n

0

0

50

100

150

p [

MeV

fm

-3]

YL = 0.4, T = 50 MeV

YL = 0.1, T = 50 MeV

Yν = 0, Τ = 0

DD-ME2

NL3

In the case of model DD-ME2 U(σ) = 0, and the couplings gσ , gω and gρ aredensity-dependent.

In the case of model NL3 U(σ) = g2σ3/3 + g3σ

4/4 6= 0, but the couplings gσ , gω andgρ are density-independent.

Results for bulk viscosity

β-equilibration rates (DD-ME2)

n p + e− + νe p + e− n + νe

1 10 100T [MeV]

10-40

10-30

10-20

10-10

10-40

10-30

10-20

10-10

Γ1

[MeV

4]

nB/n

0 = 0.5

nB/n

0 = 1

nB/n

0 = 2

nB/n

0 = 3

YL = 0.4

YL = 0.1(a)

(b)

10-12

10-10

10-8

10-6

Γ2

[MeV

4]

nB/n

0 = 0.5

nB/n

0 = 1

nB/n

0 = 3

1 10 100T [MeV]

10-10

10-8

10-6

YL = 0.4

YL = 0.1(a)

(b)

The neutron decay rate Γ1 is exponentially suppressed at low temperatures because ofdamping of anti-neutrino population in the degenerate matter.

The electron capture rate Γ2 has a finite low-temperature limit which is ∝ T3.

In the regime of interest Γ1 Γ2, therefore the electron capture process dominates inthe β-equilibration and the bulk viscosity.


β-relaxation rate (DD-ME2)

10-17

10-15

10-13

10-11

λA

[M

eV

]

YL = 0.4

YL = 0.1

Yν = 0

1 10 100T [MeV]

10-17

10-15

10-13

10-11

nB/n

0 = 3

nB/n

0 = 0.5(a)

(b)

f = 10 kHz

f = 1 kHz

The β-relaxation rate λA determines the frequency at which the bulk viscosity reaches itsresonant maximum (ωmax = λA).The relaxation rate is slowest in the neutrino-transparent case, and increases with thelepton fraction in the neutrino-trapped case.In neutrino-trapped matter λA ω for oscillation frequencies typical to neutron starmergers and supernovas ⇒ the bulk viscosity takes the form ζ ≈ C2/(λA2).The neutrino-transparent matter instead features a relaxation rate which is comparable tothe oscillation frequencies at typical temperatures T ' 2÷ 7 MeV.


The susceptibility prefactor C2/A (DD-ME2)

104

106

C2/A

[M

eV

4]

T = 5 MeVT = 50 MeV

1 2 3n

B/n

0

104

106

YL = 0.1

YL = 0.4

(a)

(b)

C =

0

C =

0

1 2 3n

B/n

0

0,188

0,19

0,192

0,194

0,16

0,17

0,18

0,19

Yp

T = 5 MeV

T = 50 MeV

YL = 0.1

(a)

(b)

nB/n

0 =

1.6

The susceptibility A does not depend strongly on the density and temperature and hasroughly the same order of magnitude A ∼ 10−3 MeV−2.The susceptibility C increases with density and at high temperatures T & 30 MeVcrosses zero at certain values of density where the proton fraction attains a minimum.At this critical density the system is scale-invariant: it can be compressed and remain inbeta equilibrium ⇒ the bulk viscosity drops to zero at the critical point.


Bulk viscosity of neutrino-trapped matter (DD-ME2)

1 10T [MeV]

1020

1022

1024

1026

ζ [

g c

m- s

-1]

nB/n

0 = 3

nB/n

0 = 2

nB/n

0 = 1

nB/n

0 = 0.5

1 10 100T [MeV]

(b)(a)

YL

= 0.1 YL

= 0.4

The density dependence of the bulk viscosity follows that of the susceptibility C2/A.The temperature dependence of ζ arises mainly from that of β-relaxation rate λA ∝ T2.Bulk viscosity is independent of oscillation fequency and decreases as ζ ∝ T−2 in theneutrino-trapped regime.This scaling breaks down at high temperatures T ≥ 30 MeV where the bulk viscosity hassharp minimums ζ → 0 when the matter becomes scale-invariant.


Bulk viscosity of neutrino-trapped matter (NL3)

1 10T [MeV]

1020

1022

1024

1026

1028

ζ [

g c

m- s

-1]

nB/n

0 = 3

nB/n

0 = 2

nB/n

0 = 1

nB/n

0 = 0.5

1 10 100T [MeV]

(b)(a)

YL

= 0.1 YL

= 0.4

The density dependence of the bulk viscosity follows that of the susceptibility C2/A.The temperature dependence of ζ arises mainly from that of β-relaxation rate λA ∝ T2.Bulk viscosity is independent of oscillation fequency and decreases as ζ ∝ T−2 in theneutrino-trapped regime.This scaling breaks down at high temperatures T ≥ 30 MeV where the bulk viscosity hassharp minimums ζ → 0 when the matter becomes scale-invariant.


Bulk viscosity of neutrino-transparent matter

1 2 4 8T [MeV]

1021

1024

1027

1030

ζ [

g c

m-1

s-1

]

nB/n

0 = 3

nB/n

0 = 2

nB/n

0 = 1

nB/n

0 = 0.5

1 2 4 8T [MeV]

Yν = 0

f = 1 kHz f = 10 kHz(a) (b)

The bulk viscosity in neutrino-transparent matter is frequency-dependent.

The relaxation rate is slower for neutrino-transparent matter, and the resonant peak of thebulk viscosity occurs within its regime of validity.

It attains its maximum value at temperature T ' 2÷ 7 MeV, where ω = λA.

This is the temperature range which is relevant for neutron-star mergers. 1

1M. G. Alford, et al., On the importance of viscous dissipation and heat conduction in binary neutron-star mergers, 2017


Bulk viscosity of neutrino-transparent matter (NL3)

1 2 4 8T [MeV]

1021

1024

1027

1030

ζ [

g c

m-1

s-1

]

nB/n

0 = 3

nB/n

0 = 2

nB/n

0 = 1

nB/n

0 = 0.5

1 2 4 8T [MeV]

Yν = 0

f = 1 kHz f = 10 kHz(a) (b)

The bulk viscosity in neutrino-transparent matter is frequency-dependent.

The relaxation rate is slower for neutrino-transparent matter, and the resonant peak of thebulk viscosity occurs within its regime of validity.

It attains its maximum value at temperature T ' 2÷ 7 MeV, where ω = λA.

This is the temperature range which is relevant for neutron-star mergers. 2

2M. G. Alford, et al., On the importance of viscous dissipation and heat conduction in binary neutron-star mergers, 2017


Bulk viscosity of baryonic matter (DD-ME2)

1 10T [MeV]

1022

1025

1028

1031

ζ [

g c

m- s

-1]

1 10 100T [MeV]

(b)(a)

f = 1 kHz f = 10 kHz

ν-trapped

ν-trapped

ν-transparent

ν-transparent

3 3

1

2 2

1

We interpolate our numerical results for the bulk viscosity between theneutrino-transparent and neutrino-trapped regimes in the interval 5 ≤ T ≤ 10 MeV.

The bulk viscosity in the neutrino transparent regime is larger, and drops by orders ofmagnitude as the matter enters the neutrino-trapped regime.

The bulk viscosity attains its maximum at temperatures T ' 2÷ 6 MeV.


Bulk viscosity of baryonic matter (NL3)

1 10T [MeV]

1022

1025

1028

1031

ζ [

g c

m- s

-1]

1 10 100T [MeV]

(b)(a)


ν-transparent

ν-trapped

ν-trapped

ν-transparent

1

2

3 3

2

1

We interpolate our numerical results for the bulk viscosity between theneutrino-transparent and neutrino-trapped regimes in the interval 5 ≤ T ≤ 10 MeV.

The bulk viscosity in the neutrino transparent regime is larger, and drops by orders ofmagnitude as the matter enters the neutrino-trapped regime.

The bulk viscosity attains its maximum at temperatures T ' 2÷ 6 MeV.


Estimation of oscillation damping timescale

The energy density of baryonic oscilations with amplitude δnB is

ε =K2

(δnB)2

nB.

Coefficient K is the compressibility of nuclear matter

K = nB∂2ε

∂n2B.

The enegy dissipation rate per volume by bulk viscosity is

dεdt

=ω2ζ

2

(δnB

nB

)2

.

The characteristic timescale required for dissipation is τ = ε/(dε/dt)

τ =KnB

ω2ζ.

In the high-frequency (ω λA), low-frequency (ω λA) limits and at the maximumof the bulk viscosity (ω = λA) we find

τhigh =KnB

λC2, τlow =

λ2A2

ω2

KnB

λC2, τmin = 2

KnB

λC2.


Nuclear compressibility for two EoS

1 2 3n

B/n

0

0

200

400

600

800

1000

K [

MeV

]

T = 1 MeV

T = 5 MeV

T = 50 MeV

DD-ME2

NL3


Oscillation damping timescale for DD-ME2 model

1 10T [MeV]

10-4

10-3

10-2

10-1

100

101

τ [

s]

1 10T [MeV]

nB/n

0 = 3

nB/n

0 = 2

nB/n

0 = 1

nB/n

0 = 0.5

(b)(a)


Damping timescales are comparable to the merging timescales τmerg ' 10 ms attemperatures T . 7 MeV where neutrinos are not trapped.Therefore, bulk viscosity will have its greatest impact on neutron star mergers in regionsthat are neutrino transparent rather than neutrino trapped.This also implies weak damping of gravitational waves emitted by the oscillations of thepost-merger remnant in the high-temperature, neutrino-trapped phase of evolution.


Oscillation damping timescale for NL3 model

1 10T [MeV]

10-4

10-2

100

102

τ [

s]

nB/n

0 = 3

nB/n

0 = 2

nB/n

0 = 1

nB/n

0 = 0.5

1 10T [MeV]

(b)(a)


Damping timescales are comparable to the merging timescales τmerg ' 10 ms attemperatures T . 7 MeV where neutrinos are not trapped.Therefore, bulk viscosity will have its greatest impact on neutron star mergers in regionsthat are neutrino transparent rather than neutrino trapped.This also implies weak damping of gravitational waves emitted by the oscillations of thepost-merger remnant in the high-temperature, neutrino-trapped phase of evolution.

Conclusions

Conclusions

We studied the bulk viscosity of dense nuclear matter in the case of trapped neutrinos.

We find that the bulk viscosity is dominated by electron capture process and its inverse.

The bulk viscosity attains its maximum at temperatures 2÷ 7 MeV forneutrino-transparent matter.

The neutrino-trapped bulk viscosity does not attain its maximum within its range ofvalidity because of faster equilibration rates.

The nuclear matter becomes scale-invariant at a critical density at high temperaturesT ≥ 30 MeV driving the bulk viscosity to zero at those points.

The bulk viscosity of neutrino-trapped matter is several orders of magnitude smaller thanthat of neutrino-transparent matter.

We find that the neutron star oscillation damping timescales in neutrino-trapped matterare likely to be too long to affect the evolution of neutron star mergers.

This also implies weak damping of gravitational waves emitted by the oscillations of thepost-merger remnant in the high-temperature, neutrino-trapped phase of evolution.

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Bulk viscosity of baryonic matter with trapped neutrinos ... · Bulk viscosity of baryonic matter with trapped neutrinos and its application to neutron star mergers Arus Harutyunyan