Constraining Gravity with the Equation of State in Neutron Stars

Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary

Constraining gravity with the equation of state in neutron stars

Felipe J. Llanes-Estrada

Universidad Complutense de Madrid Departamento de F´ısica Te´orica I

Institute of Nuclear Theory Seminar September 16th 2015, U. of Washington at Seattle

In collaboration with M. Aparicio, A. de la Cruz Dombriz, V. Zapatero (work in progress) and A. Dobado, A. Oller Phys.Rev. C85 (2012) 012801

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Constraining gravity with the equation of state in N stars

Motivation

Static neutron stars

Constraining Cavendish’s constant with heavy stars

Modiﬁed gravity

Motivation

Static neutron stars

Constraining Cavendish’s constant with heavy stars

Modiﬁed gravity

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Often emphasized: Neutron stars test structure of matter

Metallic crust

Various phases of hadron matter

Possibly a quark core

10 km 1.2 sM

# include INT TALENT 2015: Nuclear Physics of Neutron Stars and Supernovae

(hep-ph/0503184)

Inside the star ◮ ρ < 3 4ρ − 0 ◮ g = O(1012)m/s2 vs. O(300)m/s2 outside (where GR tests with pulsars are performed) or O(106)m/s2 at white dwarves.

i.e. we extrapolate General Relativity 6 orders of magnitude to learn nuclear physics only a factor 3-4 away !!??

Can nuclear physics constrain General Relativity inside a neutron star?

Dark matter is obscure. Who missed something? D. Clowe et al. Astrophys. J. Lett. 648 L109 (2006)

What about accelerated expansion, e.g. primordial inﬂation? Add incompatibility of GR with quantum theory, and prediction of spacetime singularities

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Binary neutron star systems used to constrain gravity

◮ Two-solar mass pulsar J0348+0432 with white dwarf companion ◮ Grav. wave emission constrained from dT /dt

◮ Constraint on αPSR (scalar/tensor coupling ratio), Science 340 1233232 (2010)

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Neutron star interior great for Gen. Rel. tests Large fractional binding energy

J.Antoniadis et al., Science 340, 1233232 (2013)

Left: binary merger simulation by Alan Calder (not equilibrium either). Right: theory prediction of quadrupole moment Q/J2 for three equations of state

(arXiv:1503.05405)

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Only probes of interior that might get to us are neutrinos

(arXiv:0702613)

4 ]

32 Combined 3

) [ 10 IMB + 2

N ( E 1 Kam-II

0 1

0.8 Kam-II ) + 0.6 IMB f ( E 0.4 0.2 0 12 10 8 6 t [ s ] 4 2 0 0 10 20 30 40 50 60 E+ [ MeV ] Analysis of SN87A; not a neutron star in equilibrium Resort to indirect means, such as bulk properties (M, R, T , I , τcooling . . . ) (very much like nuclear physics)

Motivation

Static neutron stars

Constraining Cavendish’s constant with heavy stars

Modiﬁed gravity

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Gravity acceleration in General Relativity (G = c = 1)

dΦ M(r) g = = dr r 2

M(r) + 4πr 3P(r) g = . r(r 2M(r)) − g O(1012)m/s2 ≃

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Gravity acceleration in General Relativity (G = c = 1)

dΦ M(r) g = = dr r 2

M(r) + 4πr 3P(r) g = . r(r 2M(r)) − g O(1012)m/s2 ≃

PRC77 065803 (2008)

◮ Hydrostatic equilibrium: pressure compensates weight of upper layers ◮ Causality limits the achievable pressure, c2 c2 = dP ≥ s dρ ◮ But nothing limits the amount of matter falling on the star ◮ Only solution: increase density

◮ When R∗ < RSchwarzschild = 2M∗, Gen. Rel. predicts collapse ◮ Finding heavier N-stars tests General Relativity

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Two-solar mass neutron star Shapiro delay: allows for absolute measurement of the mass (Demorest et al. Nature 2010)

R ∆t = log(1 nˆ nˆ ) s − − 1 · 2 c Conﬁrmed with 2nd example, Antoniadis et al. Science 340 1233232 (2013)

Newtonian Hydrostatic equilibrium

G M(r) dM dP = N − r 2 dA

General Relativistic Hydrostatic equilibrium (static, spherical body)

dP G (ε(r) + P(r))(M(r) + 4πr 3P(r)) = N . dr − r 2 1 2GN M(r) − r

Newtonian Hydrostatic equilibrium

G M(r) dM dP = N − r 2 dA

General Relativistic Hydrostatic equilibrium (static, spherical body)

dP G (ε(r) + P(r))(M(r) + 4πr 3P(r)) = N . dr − r 2 1 2GN M(r) − r

Chiral Effective Theory Free neutron gas Causality limit c

P(m 3

0 0 2 4 6 8 10 12 14 16 18 ε 4 (mπ )

So what is your crazy model of the star’s inside? F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Supplemented by equation of state

Chiral Effective Theory Free neutron gas Causality limit c

P(m 3

0 0 2 4 6 8 10 12 14 16 18 ε 4 (mπ )

The optimal “random” sphere packing fraction is about 63.5% The optimal “crystal” packing fraction is about 74% (Kepler’s conjecture)

− ◮ Nucleon radius (measured in e p scattering) rN 0.88 fm ≃ 1/3 ◮ Nuclear radius rA 1.2 fmA ≃ ◮ When ρ 133 MeV /fm3 2.8 m4 ≃ ≃ π the volume evacuated by the neutrons is important

(1.-x**2.-y**2.)**(1./2.) (1.-x**4.-y**4.)**(1./4.)

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-1 -1 -0.5 -0.5 0 0 1 1 0.5 0.5 0.5 0.5 0 0 -0.5 -0.5 -1 1 -1 1

(1.-x**8.-y**8.)**(1./8.) (1.-x**12.-y**12.)**(1./12.)

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-1 -1 -0.5 -0.5 0 0 1 1 0.5 0.5 0.5 0.5 0 0 -0.5 -0.5 -1 1 -1 1

(plotted are N=2,4,8,12)

2500 Deformed neutron Experimental neutron resonances P=+

2000 M (MeV)

1500

10000 2 4 6 8 10 12 14 16 18 20 Cubicity index N Cost of “cubing” the neutron: approximately 150 MeV

Chiral Effective Theory Free neutron gas

7 Causality limit cs

5 ) 4 4 π

P(m 3

0 0 2 4 6 8 10 12 14 16 18 ε 4 (mπ )

FLE, Moreno-Navarro Mod.Phys.Lett. A27 (2012) 1250033

Hyperons are expected at high density

400 nucleons & leptons nucleons, hyperons & leptons 2 ) 350 -3 300 1.5 (solar mass units)

250 G Hulse-Taylor PSR 200 1 150

Pressure P (MeV fm 100 0.5 (a) 50 (b)

0 0 Gravitational mass M 0 200 400 600 800 1000 1200 0 0.5 1 1.5 -3 -3 Energy density ε (MeV fm ) Central baryon number density ρ (fm )

I. Vida˜na, 1509.03587

◮ Most often, just soften ◮ Exploit it to put bounds by working from the “stiﬀest” side

Vr=1 Vr=1 Op5 Op6 Leading Order Next−to−Leading Order 1 2

... Vr=2 Op6 3.1 Next−to−Leading Order

...

3.2

(Lacour, Oller and Meissner Ann. Phys. 326 (2011) 241, consistent power counting in nuclear matter)

Symmetric Nuclear Matter 30

5 E/A (MeV) 0

-5

-10

-15

-20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 rho

(Lacour, Oller and Meissner 2009, symmetric nuclear matter Np=Nn)

Free neutron gas ChEfT, no constraint 2.5 Causal ChEfT

1.5 sun

M/M 1

0.5

0 6 8 10 12 14 16 R (km)

600

500

400

300 (MeV) F P 200

100 Free neutron gas Chiral Effective Theory 0 0 4 8 12 16 ε 4 (mπ )

100 Chiral Effective Theory 10 Free neutron gas 1 0.1 ) 4 π 0.01 P(m 0.001 0.0001 1e-05 1e-06 0 100 200 300 400 500 600 PF (MeV)

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary

Causality: cs c ≤

Causality limit cs

1.2 2 0.9 /c) s (c 0.6

0.3

0 0 4 8 12 16 ε 4 (mπ )

Motivation

Static neutron stars

Constraining Cavendish’s constant with heavy stars

Modiﬁed gravity

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary

Extensions of General Relativity predict GN (g)

For example, arXiv:0410117

GN constant r 0 ≃ → 1 q GN r r ∝ kq ∝ →∞ q 10−6 ≃ Dozens of works 0901.2963, hep-th/9504014, hep-ph/0207282, astro-ph/9501066 . . .

dP G (ε(r)+ P(r))(M(r) + 4πr 3P(r)) = N dr − r 2 1 2GN M(r) − r Where Eﬀective Theory becomes unreliable use the steepmost equation of state P = P + c2(ρ ρ ) 0 − 0 (lack of knowledge of dense QCD does not alter conclusions)

2.5

1.5 sun 0 G /G = 1 M/M 1 0 G /G = 0.92 0 G /G = 0.885 0.5

0 6 8 10 12 14 16 R (km)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 ×10 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1.4

1.2 0 N

/G 1 N G

0.8

0.6

2 g (m/s )

Dobado, LL-E and Oller PRC85 (2012) 012801

r∗ = 9.4 1.2 km ± Guillot and Rutledge APJ796,1 (2014).

Typical calculated radii with GR: r∗ 12 13km ≃ − New light U-boson exchange? (1509.02128)

1.5 sun 0 G /G = 1 M/M 1 0 G /G = 0.95 0 G /G = 0.9 0.5

0 6 8 10 12 14 16 R (km) This measurement brings tension to the ﬁeld

The three EOS of Hebeler et al., APJ 773 11 (2013)

Motivation

Static neutron stars

Constraining Cavendish’s constant with heavy stars

Modiﬁed gravity

Hu-Sawicki model that provides accelerated universe’s expansion:

b R S = R cH02 − R 1+ d cH02

with c = 6 (1 Ωm) d/b; H0 = 0.999456 ∗ − ∗ Ωm = 0.35831 d = 1, b = 209.7263765 R But many others too, Tsujikawa: R µRT tanh R − T 2 2 −n Starobinsky: R + λRS 1+ R /RS 1 − −R/RE Exponential: R βRE (1 e ) − − K. Bamba et al., 1108.2557

Hu-Sawicki model that provides accelerated universe’s expansion:

b R S = R cH02 − R 1+ d cH02

Two possibilities

◮ limR→∞ f (R)= Λ n− ◮ f (R) R→∞ αR ∼ D.Saez Gomez, 1207.5472

1 S = d4x√ g [R + f (R)] 16πG Z −

1 1 Rµν R gµν = [ 8πGTµν µ νfR − 2 1+ fR − −∇ ∇ α 1 +g fR + (f (R) RfR ) g ] µν ∇ ∇α 2 − µν

ds2 = B(r) dt2 A(r) dr 2 r 2(dθ2 + sin2 θ dφ) − −

◮ In GR’s Schwarzschild: A(r), B(r), but R =8πGT is algebraically constrained ◮ In f (R) theories, a diﬀerential equation for R: α 8πGT 2 f (R) 3 αfR R = − − ∇ ∇ 1 fR −

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Equations of hydrostatic equilibrium generalizing TOV

′ ′ ′′ ′ A B 2 A R = R [8πG(ρ 3p) (1 fR )R 2f (R)] 2A − 2B − r − 3fRR − − − − 2rA A 3B′ A′ = 8πGA(ρ + 3p)+ R 3(1 + fR ) 2 − 2rB ′ ′ A 3B 3 3B ′ fR R + + fRR R + Af (R) − 2 2rB − r 2B B′ A′ B′ 2A′B 2B B′′ = + + + [ 8πGAp 2 A B rA (1 + fR ) − ′ A B 2 ′ A R + + fRR R f (R) − 2 2B r − 2 ρ + p B′ p′ = − 2 B

◮ Unlike earlier studies, e.g. Astashenov et al. 1309.1978 we are not using perturbation theory in a ◮ System solved by 4th order Runge-Kutta ◮ Initial conditions to obtain regularity, ﬁnite pressure, matching

−6 ◮ Solar system tests: f (Rss) < 10 but what is Rss? | | −46 −2 ◮ RFRW 3 10 km ∼ × ◮ RSchwz = 0; dimensionally, Kretschmann’s scalar 2 µνρσ 12rs −17 −2 R R = 3 10 km (at r = R⊙) µνρσ r 6 ∼ × ◮ R + aR2 + O(R4) a can currently be huge µν (Note: not general-most extension, can play with indices Rµν R )

Does the burden grow quadratically with the paperwork?

http://www.freshtracks.co.uk/blog/how-to-lead-a-happy-committee/

′ (obtained by forcing A, B, B to be Schwarzschild-like at R∗)

ds2 = B(r) dt2 A(r) dr 2 r 2(dθ2 + sin2 θ dφ) − −

(It actually oscillates indeﬁnitely with a small amplitude a) ∝

(solution of the geodesic eqs. courtesy of M. Aparicio)

◮ State equations of Hebeler et al. APJ773:11 (2013) ◮ Matching to exterior Schwarzschild (careful) ◮ Systematic improvement needed: counting in the eq. of state ◮ Find heavier stars

◮ r 2 In Gen. Rel. “quantity of matter” M(r) = 4π 0 r drǫ(r) coincides with Schwarzschild’s mass R ◮ Shapiro mass, Newtonian potential at inﬁnity... 3/5 ◮ (m1+m2) In grav. wave detection, “Chirp mass” = 1/5 in M (m1+m2) general relativity; so m1, m2 are Schwarzschild masses. ◮ In modiﬁed gravity the solutions are not tagged by M(r) in the same way

Eﬀect of three-body forces from potential models

Expect repulsion, higher pressure, when including them.

S. Gandolﬁ et al., 1307.5815

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary Equation of state: learning from other groups NNLO not much better than NLO (large two-pion exchange)

20 AFDMC LO AFDMC NLO AFDMC N2LO =0.8 fm

15 0 R

10 E/N [MeV] =1.2 fm 0 R

0 0 0.05 0.1 0.15 n [fm-3]

◮ Extend dispersive analyses of the EFT one more order (with J. Oller) ◮ Systematize the matching of solutions to general relativity ◮ Explore several f (R) alternatives

◮ Neutron star properties depends on QCD equation of state ◮ 2 solar-mass star quite high: several equations of state ruled out in General Relativity ◮ From relatively safe knowledge: constrain Cavendish constant G ◮ ∆ 12% at neutron star G ≤

◮ aR2 mod. gravity: a not well constrained elsewhere ◮ With current nuclear knowledge, O(1) bound ◮ Outer metric not quite Schwarzschild, how to tag solutions with a mass precisely? ◮ Safer: provide controlled errors and counting at the expense of some “realism” in the EOS

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars Motivation Static neutron stars Constraining Cavendish’s constant with heavy stars Modiﬁed gravity Summary

Constraining gravity with the equation of state in neutron stars

Felipe J. Llanes-Estrada

Universidad Complutense de Madrid Departamento de F´ısica Te´orica I

Institute of Nuclear Theory Seminar September 16th 2015, U. of Washington at Seattle

In collaboration with M. Aparicio, A. de la Cruz Dombriz, V. Zapatero (work in progress) and A. Dobado, A. Oller Phys.Rev. C85 (2012) 012801

F. J. Llanes-Estrada Constraining gravity with the equation of state in neutron stars