Constraining the nuclear EoS by combining nuclear data and GW observations

Transcription

1 Constraining the nuclear EoS by combining nuclear data and GW observations Michael McNeil Forbes Washington State University (Pullman) and University of Washington

2 A Minimal Nuclear Energy Density Functional Bulgac, Forbes, Jin, Perez, Schunk arxiv: (accepted PRC 2018) 6-7 parameters Better accuracy than other NEDF (without beyond mean-field corrections) fewer parameters (others have 13-30) Not fully optimized Separation of dominant from subdominant parameters Masses, Radii, Separation energies, Neutron matter Suitable for dynamics (reasonably fission properties without tuning!)

3 z } { z } { Inspired by liquid drop {z } Inspired by liquid drop formula 4-5 parameters: Global mass fits (2375 nuclei) MeV compared to to 4-6 parameter Liquid Drop Formula ( MeV) Also get 883 charge radii 0.041fm SeaLL1 hydro Comments n Adjusted (see Fig. 5) a 0 0 same Insignificant b (10) 685.6(2) c c 0 n = 10m 3~2 a a 1 = n 1/3 0 b b 0 n b (61) 94.9(14) c 1 256(25) Fixed in orbital-free theory a a 2 = a n a 0 a 1 b b 2 = b n b 0 b 1 c c 2 = c n c 0 c 1 a n 32.6 same from neutron matter EoS (16) b n same from neutron matter EoS (16) c n same from neutron matter EoS (16) s 3.93(15) 3.370(50) W (52) 0.0 Fixed in orbital-free theory g N/A g 0 fit in Ref. [145] apple N/A 0.2 Semi-classical (see section III H) ~ 2 2m same units (MeV = fm = 1) e same cgs units (4 0 = 1) E even-even nuclei nuclei r charge radii charge radii Table II. Best fit parameters for the SeaLL1 functional (in bold) and the orbital-free approximation (next column in italic when di erent). The errors quoted for the fit parameters should be interpreted as estimating by how much this parameter can be independently changed while refitting the other and incurring a cost of at most E < 0.1 MeV.

4 E[n n,n p ]= + z } { h 2 2m ( n + p )+ z } { 2X a j n 5/3 + b j n 2 + c j n 7/3 j=0 2j + z } { X h 2 + s 2m k ~rn q k 2 + W 0 ~J ~rn + {z } q=n,p + X g (~r) q (~r) 2 q=n,p {z } ˆ 3 ~r 0 n p (~r)n p (~r 0 4/3 ) 2 k~r - ~r 0-3e2 np (~r) k 4 3 {z } + e2 SeaLL1 NEDF

5 E[n n,n p ]= + z } { h 2 2m ( n + p )+ z } { 2X a j n 5/3 + b j n 2 + c j n 7/3 j=0 2j + z } { X h 2 + s 2m k ~rn q k 2 + W 0 ~J ~rn + {z } q=n,p + X g (~r) q (~r) 2 q=n,p {z } ˆ 3 ~r 0 n p (~r)n p (~r 0 4/3 ) 2 k~r - ~r 0-3e2 np (~r) k 4 3 {z } + e2 SeaLL1 NEDF No effective mass good level densities Orbital-free version from semi-classical expansion (2-fluid hydrodynamics)

6 E[n n,n p ]= + + z } { h 2 2m ( n + p )+ z } { 2X a j n 5/3 + b j n 2 + c j n 7/3 j=0 z } { X h 2 s 2m k ~rn q k 2 + W 0 ~J ~rn {z } 2j + Neutron skin NEDF 0 0 K 0 S L L Pb 48 Ca [fm 3 ] [fm] [fm] SeaLL1 q=n,p X g (~r) q (~r) 2 Table III. Saturation, symmetry, and neutron skin properties for SeaLL1. q=n,p All values in MeV unless otherwise specified. {z } ˆ 3 ~r 0 n p (~r)n p (~r 0 4/3 ) 2 k~r - ~r 0-3e2 np (~r) k 4 3 {z } + e2 SeaLL1 NEDF β=(nn-np)/(nn+np) Inspired by UFG PCA reduces to 3 parameters (n0, ϵ0, S2) K0 ±25MeV and L2 not tightly constrained (one more parameter) β 4 terms fixed by neutron matter

7 E[n n,n p ]= + z } { h 2 2m ( n + p )+ z } { 2X a j n 5/3 + b j n 2 + c j n 7/3 j=0 2j + z } { X h 2 + s 2m k ~rn q k 2 + W 0 ~J ~rn + {z } q=n,p + X g (~r) q (~r) 2 q=n,p {z } ˆ 3 ~r 0 n p (~r)n p (~r 0 4/3 ) 2 k~r - ~r 0-3e2 np (~r) k 4 3 {z } + e2 SeaLL1 NEDF Adjusts surface tension, diffuseness, pairing etc. Required and constrained by mass/radii fits

8 8 6 Mean: 0.93 MeV St.dev: 1.46 MeV deformed spherical Residuals Mth Mexp [MeV] Mth Mexp [MeV] neutron number N Mean: 0.93 MeV St.dev: 1.46 MeV deformed spherical proton number Z 606 even-even nuclei: χe=1.74mev (1.46MeV w/o bias) Fit to 196 spherical nuclei 345 charge-radii χr=0.034fm No beyond mean-field corrections Good for self-consistent dynamics CoM correction for spherical nuclei: 1.54MeV 0.96MeV 2n (2p) separation energies χe=0.69mev (0.59MeV) Lower than any other NEDF!

9 Neutron Drip Line Z Z b SLy4 UNEDF1 FRLDM SeaLL1 r-process (nsm) r-process (classic) r-process (nsm-unedf1) r-process (FRDM) r-process (DZ31) N

10 Density Profiles 0.08 n(r)[fm 3 ] Pb 48 Ca r [fm] Solid from De Vries et al. At. Data. Nucl. Data Tables 36, (1987)

11 Fission (for Free) 240 Pu fission pathway Energy (MeV) E A E II E B SeaLL1 UNEDF SkM Q 20 [b] Similar 240 Pu fission behaviour as SkM* but no tuning to match fission barriers Q30 [b] 3/ Pu - SeaLL Other NEDFs have similar errors in barrier heights Q 20 [b] 0

12 Subdominant and Free Parameters to Tune Entrainment terms GDR, Thomas-Kuhn-Reiche sum rule, Gamow-Teller resonance Gradient terms/density-dependent kinetic terms Surface tension and diffuseness, Static dipole susceptibility Proton and neutron effective masses Isospin-breaking pairing/spin-orbit coupling Spin-orbit splittings, Pairing gaps Decoupled from primary mass fit parameters

13 E[n n,n p ]= + z } { h 2 2m ( n + p )+ z } { 2X a j n 5/3 + b j n 2 + c j n 7/3 j=0 2j + z } { X h 2 + s 2m k ~rn q k 2 + W 0 ~J ~rn + {z } q=n,p + X g (~r) q (~r) 2 q=n,p {z } ˆ 3 ~r 0 n p (~r)n p (~r 0 4/3 ) 2 k~r - ~r 0-3e2 np (~r) k 4 3 {z } + e2 SeaLL1 NEDF β=(nn-np)/(nn+np) Inspired by UFG PCA reduces to 3 parameters (n0, ϵ0, S2) K0 ±20MeV and L2 not tightly constrained (one more parameter) β 4 terms fixed by neutron matter

14 Nuclear EoS: L2 L E(n n,n p ) n = 0 (n)+ 2 (n) nn - n p n (n) nn - n p n 4 + Isospin expansion about symmetric nuclear matter 0 (n) =" K ( 3 ) 2 (n) =S 2 + L K ( 3 ) 4 (n) =S 4 + L K ( 3 ) = n - n 0 3n 0, n = n n + n p

15 Symmetry Properties E(n, 0) - E(n/2, n/2) n =(S 2 + S 4 + ) {z } S +(L 2 + L 4 + ) {z } L + Isospin expansion about symmetric nuclear matter L=L2 Only if (nn - np) 4 terms are insignificant See also Lattimer NPA 928 (2014) 276

16 Nuclear EoS: L2 L e(n) [MeV] n [fm] S L 2 /3n 0 (a) L/3n 0 (b) (nn-np) 2 Fit nuclei (masses, radii, skins etc.) (nn-np) 4 Seems needed for neutron matter Setting these to match neutron matter does not affect quality of fits Bulgac, Forbes, Jin, Perez, Schunk: (PRC 2018)

17 Nuclear EoS: L2 L 20 Older functionals fit QMC without 3N e(n) [MeV] n [fm] Hard to fit new QMC with (nn-np) 2 (nn-np) 4 fits neutron matter without affecting nuclear fits QMC from Wlazłowski, Holt, Moroz, Bulgac, Roche, PRL 113 (2014) [ ] L 2 /3n 0 SeaLL1 Fayans Baldo et al. QMC (2N+3N) QMC (2N)

18 L2 L 20 e(n) [MeV] 10 L 2 /3n 0 SeaLL1 Fayans Baldo et al. QMC (2N+3N) QMC (2N) n [fm]

19 SHF RMF L2 L 20 e(n) [MeV] 10 L 2 /3n 0 SeaLL1 Fayans Baldo et al. QMC (2N+3N) QMC (2N) n [fm] P.-G. Reinhard at ECT*, Trento, Italy, January, 2015, com/site/ectworkshopns2015/talks Skyrme and Relativistic Mean Field Theory Still misses neutron matter Possible that neutron matter is essentially independent of nuclei?

20 SHF RMF L2 L 20 e(n) [MeV] 10 L 2 /3n 0 SeaLL1 Fayans Baldo et al. QMC (2N+3N) QMC (2N) n [fm] P.-G. Reinhard at ECT*, Trento, Italy, January, 2015, com/site/ectworkshopns2015/talks Skyrme and Relativistic Mean Field Theory Still misses neutron matter Possible that neutron matter is essentially independent of nuclei?

21 Connect Nuclear EoS with Neutron Stars Expand from neutron matter Constrain with symmetric matter Unified EoS Compressible Liquid Drop Model (CLDM) matching Similar to Fortin et al. PRC 94 (2016) Parametrized Core Characterize the speed of sound

22 Connect Nuclear EoS with Neutron Stars Outer Crust Tabulated Data Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound with Sanjay Reddy (UW), Dake Zhou (UW), Sukanta Bose (WSU)

23 Parameters Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

24 0 Parameters Outer Crust None Outer Crust i.e. Negele-Vautherin Augment if needed to ensure convexity No parameters Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

25 2 Parameters Unified EoS Solve for Wigner-Seitz cell given homogeneous EoS Two parameters C_C Coulomb suppression sigma_delta Isospin dependence Other surface parameter fixed to smoothly match to tabulated outer crust Lattimer et al. NPA 432 (1985) 646 Steiner PRC 85 (012) Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

26 13 Parameters Pure neutron matter: 4 a, α, b, β Proton polaron: 3 µp(ñ0), up, m* Symmetric nuclear matter: 6 n0, ϵ0, K0 S2, L2, K2 Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

27 4 Parameters Start from pure Neutron matter E n (n n )= E n(n n ) n n = m n c 2 + a nn n 0 + b Generally fits QMC data and trends nn n 0 Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Energy per Neutron (MeV) L (MeV) E sym (MeV) Fermi-gas E sym = 30.5 MeV (NN) Neutron Density (fm -3 ) M (M solar ) Causality: R>2.9 (GM/c 2 ) 32 ρ central =5ρ 0 ρ central =4ρ 0 ρ central =3ρ 0 ρ central =2ρ E sym = 30.5 MeV (NN) 1.97(4) M solar M solar R (km) Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound Gandolfi et al. PRC 85 (2012) (R)

28 Expand in powers of proton fraction x p = n p n n + n p E np (n n,n p )=(1 - x p )E n (n n )+ + x p m p c 2 + p (n B ) + + (2 2 ) 2/3 2m x 5/3 p n 2/3 B + x 2 pf 2 (n B )+x 3 pf 3 (n B )+ Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

29 Bulk neutron matter E np (n n,n p )=(1 - x p )E n (n n )+ + x p m p c 2 + p (n B ) + + (2 2 ) 2/3 2m x 5/3 p n 2/3 B + x 2 pf 2 (n B )+x 3 pf 3 (n B )+ Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

30 2 Parameters Proton polaron Dilute proton gas in neutron background QMC for chemical potential Roggero et al. PRL 112 (2014) E np (n n,n p )=(1 - x p )E n (n n )+ + x p m p c 2 + p (n B ) + + (2 2 ) 2/3 2m x 5/3 p n 2/3 B + x 2 pf 2 (n B )+x 3 pf 3 (n B )+ Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

31 2 Parameters Polaron Parameters µp(ñ0) Chemical potential at saturation up Location of minimum

32 1 Parameter Proton gas Fermi energy Proton effective mass E np (n n,n p )=(1 - x p )E n (n n )+ + x p m p c 2 + p (n B ) + + (2 2 ) 2/3 2m x 5/3 p n 2/3 B + x 2 pf 2 (n B )+x 3 pf 3 (n B )+ Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

33 6 Parameters Expand and match symmetric nuclear matter 2 E(n n,n p ) nn - n p = 0 (n)+ 2 (n) n n Fix up to x 5 pf 5 (n B ) E np (n n,n p )=(1 - x p )E n (n n )+ + x p m p c 2 + p (n B ) + + (2 2 ) 2/3 2m x 5/3 p n 2/3 B + x 2 pf 2 (n B )+x 3 pf 3 (n B )+ Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

34 6 Parameters Expand and match symmetric nuclear matter 2 E(n n,n p ) nn - n p = 0 (n)+ 2 (n) n n Fix up to x 5 pf 5 (n B ) 0 (n) =" K ( 3 ) 2 (n) =S 2 + L K ( 3 ) = n - n 0 3n 0, n = n n + n p Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

35 3 Parameters Parametrize speed of sound C=cs 2 /c 2 c 2 S [c] c 2 S =1/ n [n 0 ] Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound Tews, Carlson, Gandolfi, Reddy [arxiv: ]

36 3 Parameters Parametrize speed of sound C(E)=cs 2 /c 2 Quadratic polynomial as function of energy-density (Easy to invert to get EoS) Ec Transition from homogeneous matter C max Maximum of polynomial E max Location of maximum Outer Crust None Compressible Liquid Drop Model (CLDM) Match outer crust to homogeneous matter Homogeneous Matter EoS extrapolated from neutron matter Core Parameterize speed of sound

37 Live Demos on Binder (or CoCalc, etc.) Reproducible science LIGO analysis Parameter Exploration

38 Explore Parameters

39 Principal Components Fake Data

40 Principal Components Fake Data

41 Missing? Lots! Phase transitions Clustering Pasta Finite T Out of equilibrium Core?!?

42 Conclusions New Minimal Nuclear Functional SeaLL1 Good static and dynamic properties Suggests L2 L Parametrize Neutron Star EoS from Neutron Matter Explore parameter sensitivity Share results and code