The Constant-Sound-Speed parameterization of the quark matter EoS

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1 The Constant-Sound-Speed parameterization of the quark matter EoS Prof. Mark Alford Washington University in St. Louis Alford, Han, Prakash, arxiv: Alford, Burgio, Han, Taranto, Zappalà, arxiv: Ranea-Sandoval, Han, Orsaria, Contrera, Weber, Alford, arxiv:

2 T Schematic QCD phase diagram heavy ion collider QGP hadronic gas liq non CFL CFL = color superconducting quark matter nuclear superfluid /supercond compact star µ M. Alford, K. Rajagopal, T. Schäfer, A. Schmitt, arxiv: (RMP review) A. Schmitt, arxiv: (Springer Lecture Notes)

3 Constraining QM EoS by observing M(R) What do M(R) observations tell us about hybrid stars? Lots of literature using specific models of quark matter: MIT Bag Model; (Alford, Braby, Paris, Reddy, nucl-th/ ) NJL models; (Paoli, Menezes, arxiv: ) PNJL models (Blaschke et. al, arxiv: ; Orsaria et. al.; arxiv: ) hadron-quark NLσ model (Negreiros et. al., arxiv: ) 2-loop perturbation theory (Kurkela et. al., arxiv: ) MIT bag, NJL, CDM, FCM, DSM (Burgio et. al., arxiv: ) We need a model-independent parameterization of the quark matter EoS: framework for relating different models to each other observational constraints can be expressed in universal terms

4 CSS: a fairly generic QM EoS Model-independent parameterization with Sharp 1st-order transition Constant [density-indp] ε(p) = ε trans + ε + c 2 QM (p p trans) Speed of Sound (CSS) Energy Density ε 0,QM ε trans Δε Slope = -2 c QM Quark Matter QM EoS params: p trans /ε trans ε/ε trans Nuclear Matter c 2 QM p trans Pressure Zdunik, Haensel, arxiv: ; Alford, Han, Prakash, arxiv:

5 Hybrid star M(R) Hybrid star branch in M(R) relation has 4 typical forms ε < ε crit small energy density jump at phase transition Connected M M Both R R ε > ε crit large energy density jump at phase transition M Absent Disconnected M R R

6 Phase diagram of hybrid star M(R) Δε/εtrans = λ Soft NM + CSS(c 2 QM =1) n trans/n A D B C ncausal 6.0 ε ε trans Schematic A D B C p trans/ε trans ptrans Above the red line ( ε > ε crit ), ε crit = 1 connected branch disappears ε trans p trans 2 ε trans (Seidov, 1971; Schaeffer, Zdunik, Haensel, 1983; Lindblom, gr-qc/ ) Disconnected branch exists in regions D and B. ε trans

7 Sensitivity to NM EoS and c 2 QM c 2 QM =1/3 c 2 QM =1 Δε/εtrans D NL3 B HLPS A C Δε/εtrans D NL3 B HLPS A C p trans/ε trans p trans/ε trans NM EoS (HLPS=soft, NL3=hard) does not make much difference. Higher cqm 2 favors disconnected branch.

8 Constraints on QM EoS from M max Increasing ε reduces M max Increasing p trans at first reduces then increases M max 2 M observation allows two scenarios: high p trans : very small connected branch low p trans : modest ε, no disconnected branch.

9 Low p trans and high p trans windows

10 Constraints on QM EoS from Mmax

11 Radius of heaviest star R maxm Heaviest star is typically the smallest, so lower limit on R maxm is the minimum radius of compact stars. High p trans : very short connected hybrid branch, radius like that of heaviest hadronic star. Low p trans : need to zoom in.

12 Constraints on QM EoS from RmaxM

13 Focus on low p trans and c 2 QM = 1/3 R maxm contours closely follow mass contours M max > 1.95 M requires R > km dashed line is M max = 2.1 M, requires R > 12.1 km Observation of a smaller star high transition pressure or c 2 QM > 1/3

14 Constraints on QM EoS from R1.4 M

15 Low transition pressure and R 1.4 M R 1.4 M contours roughly follow mass contours M max > 1.95 M requires R 1.4 M > 12 km (n trans n 0 ), rising with n trans. dashed line is M max = 2.1 M, requires R 1.4 M > 12.7 km Observation of a smaller 1.4 M star cqm 2 > 1/3. If p trans is high then no hybrid stars have mass 1.4 M compare Lattimer arxiv: : R > 11 km.

16 NJL models in CSS space

17 Summary of CSS CSS (Constant Speed of Sound) is a generic parameterization of the EoS close to a sharp first-order transition to quark matter. Any specific model of quark matter with such a transition corresponds to particular values of the CSS parameters (p trans /ε trans, ε/ε trans, c 2 QM ). Its predictions for hybrid star branches then follow from the generic CSS phase diagram. Existence of 2M neutron star constraint on CSS parameters. E.g., for soft NM we need cqm 2 1/3 (cqm 2 = 1/3 O(α s) in pert QCD). More measurements of M and R would strengthen the constraints. Models of quark matter tend to have cqm 2 1/3 and high transition pressure very short hybrid branch.

18 Could we identify hybrid stars via EoS? We could identify a phase transition to a high-density phase (A) Nuclear branch ends with dm/dr 0 occurs if ε/ε trans is large enough (B,D) Disconnected branch can occur with M max 2M if nuclear and quark matter are both stiff (c 2 QM 1) M M R R M R

19 Could we identify hybrid stars via EoS? We could identify a phase transition to a high-density phase (A) Nuclear branch ends with dm/dr 0 occurs if ε/ε trans is large enough (B,D) Disconnected branch can occur with M max 2M if nuclear and quark matter are both stiff (c 2 QM 1) M M R R M R We need: better measurements of M and R knowledge of nuclear matter EoS We could benefit from: theoretical constraints on parameters of QM EoS (p trans /ε trans, ε/ε trans, c 2 QM )

20 Signatures of quark matter in compact stars Observable Microphysical properties (and neutron star structure) Phases of dense matter mass, radius spindown (spin freq, age) cooling (temp, age) glitches (superfluid, crystal) Property Nuclear phase Quark phase known unknown; eqn of state ε(p) up to n sat many models bulk viscosity shear viscosity heat capacity neutrino emissivity thermal cond. shear modulus vortex pinning energy Depends on phase: n p e n p e, µ n p e, Λ, Σ n superfluid p supercond π condensate K condensate Depends on phase: unpaired CFL CFL-K 0 2SC CSL LOFF 1SC...

21 Constraints on QM EoS from Mmax

22 Density-independent c 2 QM? nucleons nucleons+kaon cond. nucleons+hyperons hybrid case I hybrid case II pure SQM P (MeV/fm 3 ) ε (MeV/fm 3 ) Kurkela, Romatschke, Vuorinen, arxiv: The perturbative QM EoS (green and yellow lines) is almost a straight line with constant c 2 QM = dp/dε at ε 300 MeV/fm3.